🧠:<think>Alright, so I have this problem about a double pendulum, and I need to derive the differential equations for it under small angle approximations. Then, I have to express the system in matrix form. Hmm, okay, let me think about how to approach this.First, I remember that a double pendulum is a system with two masses connected by rods, and it's more complex than a single pendulum because of the coupling between the two masses. Since it's a mechanical system, I should probably use Lagrangian mechanics to derive the equations of motion. That usually involves finding the kinetic and potential energies of the system.Let me start by defining the coordinates. Each pendulum has a length ( l ) and mass ( m ). The angles from the vertical are ( theta_1 ) and ( theta_2 ). So, the positions of the masses can be described using these angles.For the first mass, its coordinates are straightforward. It's just ( x_1 = l sin theta_1 ) and ( y_1 = -l cos theta_1 ). The second mass is a bit trickier because it's connected to the first. So, its position depends on both ( theta_1 ) and ( theta_2 ). Therefore, ( x_2 = l sin theta_1 + l sin theta_2 ) and ( y_2 = -l cos theta_1 - l cos theta_2 ).Now, I need to find the velocities of each mass to compute the kinetic energy. Velocity is the derivative of position with respect to time. So, let's compute ( dot{x}_1 ), ( dot{y}_1 ), ( dot{x}_2 ), and ( dot{y}_2 ).Starting with the first mass:- ( dot{x}_1 = l cos theta_1 dot{theta}_1 )- ( dot{y}_1 = l sin theta_1 dot{theta}_1 )For the second mass, it's a bit more involved because both angles are changing:- ( dot{x}_2 = l cos theta_1 dot{theta}_1 + l cos theta_2 dot{theta}_2 )- ( dot{y}_2 = l sin theta_1 dot{theta}_1 + l sin theta_2 dot{theta}_2 )Now, the kinetic energy ( T ) is the sum of the kinetic energies of both masses. Each kinetic energy is ( frac{1}{2} m v^2 ), so:( T = frac{1}{2} m (dot{x}_1^2 + dot{y}_1^2) + frac{1}{2} m (dot{x}_2^2 + dot{y}_2^2) )Plugging in the expressions for velocities:( T = frac{1}{2} m [ (l cos theta_1 dot{theta}_1)^2 + (l sin theta_1 dot{theta}_1)^2 ] + frac{1}{2} m [ (l cos theta_1 dot{theta}_1 + l cos theta_2 dot{theta}_2)^2 + (l sin theta_1 dot{theta}_1 + l sin theta_2 dot{theta}_2)^2 ] )Simplifying the first term:( (l cos theta_1 dot{theta}_1)^2 + (l sin theta_1 dot{theta}_1)^2 = l^2 dot{theta}_1^2 (cos^2 theta_1 + sin^2 theta_1) = l^2 dot{theta}_1^2 )So, the first part becomes ( frac{1}{2} m l^2 dot{theta}_1^2 ).For the second mass, expanding the squares:( (l cos theta_1 dot{theta}_1 + l cos theta_2 dot{theta}_2)^2 = l^2 cos^2 theta_1 dot{theta}_1^2 + 2 l^2 cos theta_1 cos theta_2 dot{theta}_1 dot{theta}_2 + l^2 cos^2 theta_2 dot{theta}_2^2 )Similarly, for the y-component:( (l sin theta_1 dot{theta}_1 + l sin theta_2 dot{theta}_2)^2 = l^2 sin^2 theta_1 dot{theta}_1^2 + 2 l^2 sin theta_1 sin theta_2 dot{theta}_1 dot{theta}_2 + l^2 sin^2 theta_2 dot{theta}_2^2 )Adding these together:( l^2 dot{theta}_1^2 (cos^2 theta_1 + sin^2 theta_1) + l^2 dot{theta}_2^2 (cos^2 theta_2 + sin^2 theta_2) + 2 l^2 dot{theta}_1 dot{theta}_2 (cos theta_1 cos theta_2 + sin theta_1 sin theta_2) )Simplify using ( cos^2 + sin^2 = 1 ):( l^2 dot{theta}_1^2 + l^2 dot{theta}_2^2 + 2 l^2 dot{theta}_1 dot{theta}_2 cos(theta_1 - theta_2) )So, the kinetic energy for the second mass is ( frac{1}{2} m [ l^2 dot{theta}_1^2 + l^2 dot{theta}_2^2 + 2 l^2 dot{theta}_1 dot{theta}_2 cos(theta_1 - theta_2) ] ).Therefore, the total kinetic energy ( T ) is:( T = frac{1}{2} m l^2 dot{theta}_1^2 + frac{1}{2} m [ l^2 dot{theta}_1^2 + l^2 dot{theta}_2^2 + 2 l^2 dot{theta}_1 dot{theta}_2 cos(theta_1 - theta_2) ] )Simplify this:( T = frac{1}{2} m l^2 dot{theta}_1^2 + frac{1}{2} m l^2 dot{theta}_1^2 + frac{1}{2} m l^2 dot{theta}_2^2 + m l^2 dot{theta}_1 dot{theta}_2 cos(theta_1 - theta_2) )Combine like terms:( T = m l^2 dot{theta}_1^2 + frac{1}{2} m l^2 dot{theta}_2^2 + m l^2 dot{theta}_1 dot{theta}_2 cos(theta_1 - theta_2) )Okay, that's the kinetic energy. Now, the potential energy ( V ) is due to gravity. Each mass has potential energy ( m g y ), so:( V = m g y_1 + m g y_2 = m g (-l cos theta_1) + m g (-l cos theta_1 - l cos theta_2) )Simplify:( V = -m g l cos theta_1 - m g l cos theta_1 - m g l cos theta_2 )Combine like terms:( V = -2 m g l cos theta_1 - m g l cos theta_2 )So, the Lagrangian ( mathcal{L} = T - V ):( mathcal{L} = m l^2 dot{theta}_1^2 + frac{1}{2} m l^2 dot{theta}_2^2 + m l^2 dot{theta}_1 dot{theta}_2 cos(theta_1 - theta_2) + 2 m g l cos theta_1 + m g l cos theta_2 )Now, to find the equations of motion, I need to apply the Euler-Lagrange equations for each generalized coordinate ( theta_1 ) and ( theta_2 ).The Euler-Lagrange equation is:( frac{d}{dt} left( frac{partial mathcal{L}}{partial dot{theta}_i} right) - frac{partial mathcal{L}}{partial theta_i} = 0 ) for ( i = 1, 2 )Let's compute this for ( theta_1 ) first.Compute ( frac{partial mathcal{L}}{partial dot{theta}_1} ):From ( mathcal{L} ), the terms involving ( dot{theta}_1 ) are:- ( m l^2 dot{theta}_1^2 ): derivative is ( 2 m l^2 dot{theta}_1 )- ( m l^2 dot{theta}_1 dot{theta}_2 cos(theta_1 - theta_2) ): derivative is ( m l^2 dot{theta}_2 cos(theta_1 - theta_2) )So, total:( frac{partial mathcal{L}}{partial dot{theta}_1} = 2 m l^2 dot{theta}_1 + m l^2 dot{theta}_2 cos(theta_1 - theta_2) )Now, take the time derivative:( frac{d}{dt} left( 2 m l^2 dot{theta}_1 + m l^2 dot{theta}_2 cos(theta_1 - theta_2) right) )Compute each term:- ( frac{d}{dt} (2 m l^2 dot{theta}_1) = 2 m l^2 ddot{theta}_1 )- ( frac{d}{dt} (m l^2 dot{theta}_2 cos(theta_1 - theta_2)) )Use product rule:( m l^2 [ ddot{theta}_2 cos(theta_1 - theta_2) - dot{theta}_2 sin(theta_1 - theta_2) (dot{theta}_1 - dot{theta}_2) ] )So, putting it together:( 2 m l^2 ddot{theta}_1 + m l^2 ddot{theta}_2 cos(theta_1 - theta_2) - m l^2 dot{theta}_2 sin(theta_1 - theta_2) (dot{theta}_1 - dot{theta}_2) )Now, compute ( frac{partial mathcal{L}}{partial theta_1} ):Looking at ( mathcal{L} ), the terms involving ( theta_1 ) are:- ( m l^2 dot{theta}_1 dot{theta}_2 cos(theta_1 - theta_2) ): derivative is ( -m l^2 dot{theta}_1 dot{theta}_2 sin(theta_1 - theta_2) )- ( 2 m g l cos theta_1 ): derivative is ( -2 m g l sin theta_1 )So,( frac{partial mathcal{L}}{partial theta_1} = -m l^2 dot{theta}_1 dot{theta}_2 sin(theta_1 - theta_2) - 2 m g l sin theta_1 )Putting it all into the Euler-Lagrange equation:( 2 m l^2 ddot{theta}_1 + m l^2 ddot{theta}_2 cos(theta_1 - theta_2) - m l^2 dot{theta}_2 sin(theta_1 - theta_2) (dot{theta}_1 - dot{theta}_2) - [ -m l^2 dot{theta}_1 dot{theta}_2 sin(theta_1 - theta_2) - 2 m g l sin theta_1 ] = 0 )Simplify term by term:First term: ( 2 m l^2 ddot{theta}_1 )Second term: ( m l^2 ddot{theta}_2 cos(theta_1 - theta_2) )Third term: ( - m l^2 dot{theta}_2 sin(theta_1 - theta_2) (dot{theta}_1 - dot{theta}_2) )Fourth term: ( + m l^2 dot{theta}_1 dot{theta}_2 sin(theta_1 - theta_2) )Fifth term: ( + 2 m g l sin theta_1 )So, combining the third and fourth terms:- ( - m l^2 dot{theta}_2 sin(theta_1 - theta_2) dot{theta}_1 + m l^2 dot{theta}_2^2 sin(theta_1 - theta_2) + m l^2 dot{theta}_1 dot{theta}_2 sin(theta_1 - theta_2) )Notice that the first and third terms cancel each other:( - m l^2 dot{theta}_1 dot{theta}_2 sin(theta_1 - theta_2) + m l^2 dot{theta}_1 dot{theta}_2 sin(theta_1 - theta_2) = 0 )So, only the second term remains:( + m l^2 dot{theta}_2^2 sin(theta_1 - theta_2) )Therefore, the equation becomes:( 2 m l^2 ddot{theta}_1 + m l^2 ddot{theta}_2 cos(theta_1 - theta_2) + m l^2 dot{theta}_2^2 sin(theta_1 - theta_2) + 2 m g l sin theta_1 = 0 )Hmm, that's the equation for ( theta_1 ). Now, let's do the same for ( theta_2 ).Compute ( frac{partial mathcal{L}}{partial dot{theta}_2} ):From ( mathcal{L} ), the terms involving ( dot{theta}_2 ) are:- ( frac{1}{2} m l^2 dot{theta}_2^2 ): derivative is ( m l^2 dot{theta}_2 )- ( m l^2 dot{theta}_1 dot{theta}_2 cos(theta_1 - theta_2) ): derivative is ( m l^2 dot{theta}_1 cos(theta_1 - theta_2) )So,( frac{partial mathcal{L}}{partial dot{theta}_2} = m l^2 dot{theta}_2 + m l^2 dot{theta}_1 cos(theta_1 - theta_2) )Take the time derivative:( frac{d}{dt} left( m l^2 dot{theta}_2 + m l^2 dot{theta}_1 cos(theta_1 - theta_2) right) )Compute each term:- ( frac{d}{dt} (m l^2 dot{theta}_2) = m l^2 ddot{theta}_2 )- ( frac{d}{dt} (m l^2 dot{theta}_1 cos(theta_1 - theta_2)) )Use product rule:( m l^2 [ ddot{theta}_1 cos(theta_1 - theta_2) - dot{theta}_1 sin(theta_1 - theta_2) (dot{theta}_1 - dot{theta}_2) ] )So, putting it together:( m l^2 ddot{theta}_2 + m l^2 ddot{theta}_1 cos(theta_1 - theta_2) - m l^2 dot{theta}_1 sin(theta_1 - theta_2) (dot{theta}_1 - dot{theta}_2) )Now, compute ( frac{partial mathcal{L}}{partial theta_2} ):Looking at ( mathcal{L} ), the terms involving ( theta_2 ) are:- ( m l^2 dot{theta}_1 dot{theta}_2 cos(theta_1 - theta_2) ): derivative is ( m l^2 dot{theta}_1 dot{theta}_2 sin(theta_1 - theta_2) )- ( m g l cos theta_2 ): derivative is ( -m g l sin theta_2 )So,( frac{partial mathcal{L}}{partial theta_2} = m l^2 dot{theta}_1 dot{theta}_2 sin(theta_1 - theta_2) - m g l sin theta_2 )Putting it all into the Euler-Lagrange equation:( m l^2 ddot{theta}_2 + m l^2 ddot{theta}_1 cos(theta_1 - theta_2) - m l^2 dot{theta}_1 sin(theta_1 - theta_2) (dot{theta}_1 - dot{theta}_2) - [ m l^2 dot{theta}_1 dot{theta}_2 sin(theta_1 - theta_2) - m g l sin theta_2 ] = 0 )Simplify term by term:First term: ( m l^2 ddot{theta}_2 )Second term: ( m l^2 ddot{theta}_1 cos(theta_1 - theta_2) )Third term: ( - m l^2 dot{theta}_1 sin(theta_1 - theta_2) (dot{theta}_1 - dot{theta}_2) )Fourth term: ( - m l^2 dot{theta}_1 dot{theta}_2 sin(theta_1 - theta_2) )Fifth term: ( + m g l sin theta_2 )Now, let's expand the third term:( - m l^2 dot{theta}_1 sin(theta_1 - theta_2) dot{theta}_1 + m l^2 dot{theta}_1 sin(theta_1 - theta_2) dot{theta}_2 )Which simplifies to:( - m l^2 dot{theta}_1^2 sin(theta_1 - theta_2) + m l^2 dot{theta}_1 dot{theta}_2 sin(theta_1 - theta_2) )Now, combining the third term with the fourth term:( - m l^2 dot{theta}_1^2 sin(theta_1 - theta_2) + m l^2 dot{theta}_1 dot{theta}_2 sin(theta_1 - theta_2) - m l^2 dot{theta}_1 dot{theta}_2 sin(theta_1 - theta_2) )The second and third terms cancel each other:( m l^2 dot{theta}_1 dot{theta}_2 sin(theta_1 - theta_2) - m l^2 dot{theta}_1 dot{theta}_2 sin(theta_1 - theta_2) = 0 )So, only the first term remains:( - m l^2 dot{theta}_1^2 sin(theta_1 - theta_2) )Therefore, the equation becomes:( m l^2 ddot{theta}_2 + m l^2 ddot{theta}_1 cos(theta_1 - theta_2) - m l^2 dot{theta}_1^2 sin(theta_1 - theta_2) + m g l sin theta_2 = 0 )So, now I have two equations:1. ( 2 m l^2 ddot{theta}_1 + m l^2 ddot{theta}_2 cos(theta_1 - theta_2) + m l^2 dot{theta}_2^2 sin(theta_1 - theta_2) + 2 m g l sin theta_1 = 0 )2. ( m l^2 ddot{theta}_2 + m l^2 ddot{theta}_1 cos(theta_1 - theta_2) - m l^2 dot{theta}_1^2 sin(theta_1 - theta_2) + m g l sin theta_2 = 0 )These are the equations of motion for the double pendulum. However, they are nonlinear because of the ( sin(theta_1 - theta_2) ) and ( cos(theta_1 - theta_2) ) terms, as well as the squared angular velocities.But the problem asks for the small angle approximation. So, I can linearize these equations by assuming ( theta_1 ) and ( theta_2 ) are small, meaning ( theta_1 approx 0 ), ( theta_2 approx 0 ), and their differences are also small.Under small angles, we can approximate:- ( sin theta approx theta )- ( cos theta approx 1 )Also, ( sin(theta_1 - theta_2) approx theta_1 - theta_2 ) and ( cos(theta_1 - theta_2) approx 1 ).Additionally, the squared terms like ( dot{theta}_1^2 ) and ( dot{theta}_2^2 ) can be neglected because they are higher-order small terms.So, applying these approximations to the equations:Starting with the first equation:1. ( 2 m l^2 ddot{theta}_1 + m l^2 ddot{theta}_2 (1) + m l^2 dot{theta}_2^2 (theta_1 - theta_2) + 2 m g l theta_1 approx 0 )But since ( dot{theta}_2^2 (theta_1 - theta_2) ) is a higher-order term, we can neglect it.So, equation 1 becomes:( 2 m l^2 ddot{theta}_1 + m l^2 ddot{theta}_2 + 2 m g l theta_1 = 0 )Similarly, for the second equation:2. ( m l^2 ddot{theta}_2 + m l^2 ddot{theta}_1 (1) - m l^2 dot{theta}_1^2 (theta_1 - theta_2) + m g l theta_2 approx 0 )Again, neglecting the higher-order term ( dot{theta}_1^2 (theta_1 - theta_2) ), we get:( m l^2 ddot{theta}_2 + m l^2 ddot{theta}_1 + m g l theta_2 = 0 )So, now we have two linearized equations:1. ( 2 m l^2 ddot{theta}_1 + m l^2 ddot{theta}_2 + 2 m g l theta_1 = 0 )2. ( m l^2 ddot{theta}_1 + m l^2 ddot{theta}_2 + m g l theta_2 = 0 )Let me write them more neatly:Equation 1:( 2 ddot{theta}_1 + ddot{theta}_2 + frac{2 g}{l} theta_1 = 0 )Equation 2:( ddot{theta}_1 + ddot{theta}_2 + frac{g}{l} theta_2 = 0 )Wait, let me check the coefficients. Dividing both equations by ( m l^2 ):Equation 1:( 2 ddot{theta}_1 + ddot{theta}_2 + frac{2 g}{l} theta_1 = 0 )Equation 2:( ddot{theta}_1 + ddot{theta}_2 + frac{g}{l} theta_2 = 0 )Yes, that's correct.Now, to express this system in matrix form ( mathbf{M}ddot{mathbf{Theta}} + mathbf{C}dot{mathbf{Theta}} + mathbf{K}mathbf{Theta} = mathbf{0} ).But looking at the equations, there are no damping terms (no ( dot{theta} ) terms), so matrix ( mathbf{C} ) will be zero. Also, the equations are already in terms of ( ddot{theta} ) and ( theta ), so let's arrange them accordingly.Let me write the equations:1. ( 2 ddot{theta}_1 + ddot{theta}_2 + frac{2 g}{l} theta_1 = 0 )2. ( ddot{theta}_1 + ddot{theta}_2 + frac{g}{l} theta_2 = 0 )Let me express this in matrix form. Let ( mathbf{Theta} = [theta_1, theta_2]^T ), so ( ddot{mathbf{Theta}} = [ddot{theta}_1, ddot{theta}_2]^T ).The equations can be written as:( begin{bmatrix} 2 & 1 1 & 1 end{bmatrix} begin{bmatrix} ddot{theta}_1 ddot{theta}_2 end{bmatrix} + begin{bmatrix} frac{2 g}{l} & 0 0 & frac{g}{l} end{bmatrix} begin{bmatrix} theta_1 theta_2 end{bmatrix} = begin{bmatrix} 0 0 end{bmatrix} )So, comparing with ( mathbf{M}ddot{mathbf{Theta}} + mathbf{C}dot{mathbf{Theta}} + mathbf{K}mathbf{Theta} = mathbf{0} ), we have:- ( mathbf{M} = begin{bmatrix} 2 & 1 1 & 1 end{bmatrix} )- ( mathbf{C} = begin{bmatrix} 0 & 0 0 & 0 end{bmatrix} ) (since there are no damping terms)- ( mathbf{K} = begin{bmatrix} frac{2 g}{l} & 0 0 & frac{g}{l} end{bmatrix} )Wait, but hold on. In the standard form, the mass matrix ( mathbf{M} ) is usually associated with the coefficients of ( ddot{mathbf{Theta}} ), the damping matrix ( mathbf{C} ) with ( dot{mathbf{Theta}} ), and the stiffness matrix ( mathbf{K} ) with ( mathbf{Theta} ).In our case, since there are no damping terms, ( mathbf{C} ) is indeed zero. So, the system is undamped.But let me double-check the equations. Equation 1 is:( 2 ddot{theta}_1 + ddot{theta}_2 + frac{2 g}{l} theta_1 = 0 )Equation 2 is:( ddot{theta}_1 + ddot{theta}_2 + frac{g}{l} theta_2 = 0 )So, arranging them:Equation 1: ( 2 ddot{theta}_1 + ddot{theta}_2 = - frac{2 g}{l} theta_1 )Equation 2: ( ddot{theta}_1 + ddot{theta}_2 = - frac{g}{l} theta_2 )So, in matrix form:( begin{bmatrix} 2 & 1 1 & 1 end{bmatrix} begin{bmatrix} ddot{theta}_1 ddot{theta}_2 end{bmatrix} = - begin{bmatrix} frac{2 g}{l} & 0 0 & frac{g}{l} end{bmatrix} begin{bmatrix} theta_1 theta_2 end{bmatrix} )Which is equivalent to:( mathbf{M} ddot{mathbf{Theta}} + mathbf{K} mathbf{Theta} = mathbf{0} )Since ( mathbf{C} ) is zero.Therefore, the matrices are:- ( mathbf{M} = begin{bmatrix} 2 & 1 1 & 1 end{bmatrix} )- ( mathbf{C} = begin{bmatrix} 0 & 0 0 & 0 end{bmatrix} )- ( mathbf{K} = begin{bmatrix} frac{2 g}{l} & 0 0 & frac{g}{l} end{bmatrix} )Now, interpreting these matrices:- ( mathbf{M} ) is the mass matrix. It represents the distribution of mass in the system. The diagonal terms represent the inertia of each mass, and the off-diagonal terms (1 in this case) represent the coupling between the two masses. Since the second pendulum is connected to the first, their motions are coupled, leading to the off-diagonal terms.- ( mathbf{C} ) is the damping matrix. In this case, it's zero because there's no damping in the system. If there were friction or air resistance, this matrix would have non-zero entries representing the energy dissipation.- ( mathbf{K} ) is the stiffness matrix. It represents the restoring forces in the system. The diagonal terms correspond to the stiffness (or restoring force coefficients) for each angle, and the off-diagonal terms are zero here because, under small angle approximations, the coupling in the stiffness is negligible or already accounted for in the mass matrix.Wait, actually, in the linearized system, the coupling in the stiffness might not be zero. Hmm, let me think. In the original equations, after linearization, the coupling only appears in the mass matrix because the potential energy terms are diagonal. So, the stiffness matrix remains diagonal because each angle's restoring force depends only on itself, not on the other angle. That makes sense because the potential energy terms were ( 2 m g l cos theta_1 ) and ( m g l cos theta_2 ), which after differentiation give terms proportional to ( theta_1 ) and ( theta_2 ) respectively. So, the stiffness matrix is diagonal, and the coupling is entirely in the mass matrix due to the kinetic energy terms.So, in summary, the mass matrix captures the coupling between the two masses due to their motion, while the stiffness matrix captures the individual restoring forces for each pendulum. The damping matrix is zero because there's no damping in the system.I think that's about it. Let me just recap:1. Derived the equations of motion using Lagrangian mechanics, considering kinetic and potential energies.2. Applied small angle approximations to linearize the equations.3. Expressed the resulting system in matrix form, identifying the mass, damping, and stiffness matrices.I don't see any mistakes in the steps, but let me just verify the final matrix form.Given the equations:1. ( 2 ddot{theta}_1 + ddot{theta}_2 + frac{2 g}{l} theta_1 = 0 )2. ( ddot{theta}_1 + ddot{theta}_2 + frac{g}{l} theta_2 = 0 )Yes, writing them as:( begin{bmatrix} 2 & 1 1 & 1 end{bmatrix} begin{bmatrix} ddot{theta}_1 ddot{theta}_2 end{bmatrix} + begin{bmatrix} frac{2 g}{l} & 0 0 & frac{g}{l} end{bmatrix} begin{bmatrix} theta_1 theta_2 end{bmatrix} = begin{bmatrix} 0 0 end{bmatrix} )Which is correct. So, the matrices are as I identified.Final Answer1. The set of coupled second-order differential equations under small angle approximations are:[boxed{begin{cases}2 ddot{theta}_1 + ddot{theta}_2 + dfrac{2g}{l} theta_1 = 0 ddot{theta}_1 + ddot{theta}_2 + dfrac{g}{l} theta_2 = 0end{cases}}]2. The system can be expressed in matrix form as:[boxed{begin{bmatrix} 2 & 1 1 & 1 end{bmatrix} begin{bmatrix} ddot{theta}_1 ddot{theta}_2 end{bmatrix} + begin{bmatrix} 0 & 0 0 & 0 end{bmatrix} begin{bmatrix} dot{theta}_1 dot{theta}_2 end{bmatrix} + begin{bmatrix} dfrac{2g}{l} & 0 0 & dfrac{g}{l} end{bmatrix} begin{bmatrix} theta_1 theta_2 end{bmatrix} = begin{bmatrix} 0 0 end{bmatrix}}]Where the matrices are:- Mass matrix ( mathbf{M} = begin{bmatrix} 2 & 1 1 & 1 end{bmatrix} )- Damping matrix ( mathbf{C} = begin{bmatrix} 0 & 0 0 & 0 end{bmatrix} )- Stiffness matrix ( mathbf{K} = begin{bmatrix} dfrac{2g}{l} & 0 0 & dfrac{g}{l} end{bmatrix} )The mass matrix represents the inertia and coupling between the pendulums, the damping matrix is zero as there is no damping, and the stiffness matrix represents the restoring forces due to gravity.</think>

🧠:<think>Okay, so I have two problems to solve here, both related to sequences. Let me take them one at a time.Starting with the first problem: It's about an arithmetic sequence. The first term is given as (a_1 = 5). The common difference changes every week according to the function (d(n) = n^2 - 3n + 2), where (n) is the week number starting from 1. I need to find the 10th term of the sequence presented in the 4th week.Hmm, wait, let me parse that again. So, each week, the common difference changes. So, for week 1, the common difference is (d(1)), for week 2, it's (d(2)), and so on. But the problem is asking for the 10th term of the sequence presented in the 4th week. So, does that mean that in the 4th week, the sequence has a common difference (d(4)), and we need to find the 10th term of that week's sequence?Wait, no, maybe I need to clarify. Is the sequence presented each week an arithmetic sequence with a common difference that changes each week? Or is it that the overall sequence is arithmetic, but the common difference changes each week? That doesn't quite make sense because an arithmetic sequence has a constant common difference.Wait, perhaps it's that each week, a new arithmetic sequence is presented, each with a different common difference. So, in week 1, the sequence has common difference (d(1)), week 2 has (d(2)), etc. So, in week 4, the common difference is (d(4)), and we need to find the 10th term of that week's sequence.Yes, that must be it. So, each week, the novelist presents a new arithmetic sequence starting from (a_1 = 5), but with a different common difference each week. So, in week 4, the common difference is (d(4)), and we need the 10th term of that sequence.Alright, so let me compute (d(4)) first.Given (d(n) = n^2 - 3n + 2), so plugging in (n=4):(d(4) = 4^2 - 3*4 + 2 = 16 - 12 + 2 = 6).So, the common difference for week 4 is 6.Now, the arithmetic sequence in week 4 starts at (a_1 = 5) with a common difference of 6. So, the nth term of an arithmetic sequence is given by:(a_n = a_1 + (n - 1)d)So, the 10th term would be:(a_{10} = 5 + (10 - 1)*6 = 5 + 9*6 = 5 + 54 = 59).Wait, that seems straightforward. So, the 10th term is 59. Let me just double-check.Compute (d(4)):4 squared is 16, minus 3*4 is 12, so 16 - 12 is 4, plus 2 is 6. Correct.Then, arithmetic sequence starting at 5, difference 6. 10th term: 5 + 9*6. 9*6 is 54, plus 5 is 59. Yep, that seems right.Moving on to the second problem: It's about a geometric sequence linked to the Fibonacci sequence. The first term is (b_1 = F_5), where (F_n) is the nth Fibonacci number. The common ratio is the reciprocal of the golden ratio, (phi = frac{1 + sqrt{5}}{2}). We need to compute the 5th term of this geometric sequence.Alright, let's break this down. First, I need to find (F_5). The Fibonacci sequence starts with (F_1 = 1), (F_2 = 1), and each subsequent term is the sum of the two preceding ones.So, let's compute (F_1) through (F_5):- (F_1 = 1)- (F_2 = 1)- (F_3 = F_2 + F_1 = 1 + 1 = 2)- (F_4 = F_3 + F_2 = 2 + 1 = 3)- (F_5 = F_4 + F_3 = 3 + 2 = 5)So, (b_1 = F_5 = 5).The common ratio is the reciprocal of the golden ratio. The golden ratio (phi) is (frac{1 + sqrt{5}}{2}), so its reciprocal is (frac{2}{1 + sqrt{5}}). Let me rationalize that denominator to make it easier to work with.Multiply numerator and denominator by (1 - sqrt{5}):(frac{2}{1 + sqrt{5}} * frac{1 - sqrt{5}}{1 - sqrt{5}} = frac{2(1 - sqrt{5})}{1 - 5} = frac{2(1 - sqrt{5})}{-4} = frac{-2(1 - sqrt{5})}{4} = frac{-1 + sqrt{5}}{2}).So, the common ratio (r = frac{sqrt{5} - 1}{2}).Alternatively, since (phi = frac{1 + sqrt{5}}{2}), then (1/phi = frac{sqrt{5} - 1}{2}), which is approximately 0.618.So, the geometric sequence has first term 5 and common ratio (frac{sqrt{5} - 1}{2}). We need the 5th term.The nth term of a geometric sequence is given by (b_n = b_1 * r^{n-1}).So, the 5th term is:(b_5 = 5 * left( frac{sqrt{5} - 1}{2} right)^{4}).Hmm, that exponent is 4, which might be a bit tedious, but let's compute it step by step.First, let me compute (r = frac{sqrt{5} - 1}{2}). Let me denote this as (r).Compute (r^2):(r^2 = left( frac{sqrt{5} - 1}{2} right)^2 = frac{(sqrt{5} - 1)^2}{4} = frac{5 - 2sqrt{5} + 1}{4} = frac{6 - 2sqrt{5}}{4} = frac{3 - sqrt{5}}{2}).Then, (r^3 = r^2 * r = frac{3 - sqrt{5}}{2} * frac{sqrt{5} - 1}{2}).Let me compute that:Multiply numerator: ((3 - sqrt{5})(sqrt{5} - 1)).= (3*sqrt{5} - 3*1 - sqrt{5}*sqrt{5} + sqrt{5}*1)= (3sqrt{5} - 3 - 5 + sqrt{5})= ( (3sqrt{5} + sqrt{5}) - (3 + 5) )= (4sqrt{5} - 8)So, numerator is (4sqrt{5} - 8), denominator is 4 (since 2*2=4).Thus, (r^3 = frac{4sqrt{5} - 8}{4} = sqrt{5} - 2).Now, compute (r^4 = r^3 * r = (sqrt{5} - 2) * frac{sqrt{5} - 1}{2}).Multiply numerator:((sqrt{5} - 2)(sqrt{5} - 1))= (sqrt{5}*sqrt{5} - sqrt{5}*1 - 2*sqrt{5} + 2*1)= (5 - sqrt{5} - 2sqrt{5} + 2)= (5 + 2 - 3sqrt{5})= (7 - 3sqrt{5})So, numerator is (7 - 3sqrt{5}), denominator is 2.Thus, (r^4 = frac{7 - 3sqrt{5}}{2}).Therefore, the 5th term is:(b_5 = 5 * frac{7 - 3sqrt{5}}{2} = frac{35 - 15sqrt{5}}{2}).Hmm, that's the exact value. Alternatively, if needed, we can write it as ( frac{35}{2} - frac{15sqrt{5}}{2} ), but I think the fraction form is acceptable.Let me just verify my calculations step by step to make sure I didn't make a mistake.First, (F_5 = 5), correct.Common ratio (r = 1/phi = (sqrt{5} - 1)/2), correct.Compute (r^2):((sqrt{5} - 1)^2 = 5 - 2sqrt{5} + 1 = 6 - 2sqrt{5}), divided by 4, so ( (6 - 2sqrt{5})/4 = (3 - sqrt{5})/2 ), correct.Then (r^3 = r^2 * r = (3 - sqrt{5})/2 * (sqrt{5} - 1)/2).Multiply numerators:3*sqrt(5) - 3*1 - sqrt(5)*sqrt(5) + sqrt(5)*1= 3sqrt(5) - 3 - 5 + sqrt(5)= 4sqrt(5) - 8, correct.Divide by 4: (4sqrt(5) - 8)/4 = sqrt(5) - 2, correct.Then (r^4 = r^3 * r = (sqrt(5) - 2) * (sqrt(5) - 1)/2).Multiply numerators:sqrt(5)*sqrt(5) = 5sqrt(5)*(-1) = -sqrt(5)-2*sqrt(5) = -2sqrt(5)-2*(-1) = +2So, 5 - sqrt(5) - 2sqrt(5) + 2 = 7 - 3sqrt(5), correct.Divide by 2: (7 - 3sqrt(5))/2, correct.Multiply by 5: 5*(7 - 3sqrt(5))/2 = (35 - 15sqrt(5))/2, correct.So, the 5th term is (frac{35 - 15sqrt{5}}{2}).Alternatively, if we want to write it as a decimal, we can approximate sqrt(5) ≈ 2.236.So, 15sqrt(5) ≈ 15*2.236 ≈ 33.54So, 35 - 33.54 ≈ 1.46Divide by 2: ≈ 0.73So, approximately 0.73, but since the problem doesn't specify, the exact form is better.So, summarizing:1. The 10th term of the arithmetic sequence in week 4 is 59.2. The 5th term of the geometric sequence is (frac{35 - 15sqrt{5}}{2}).I think that's it.Final Answer1. The 10th term is boxed{59}.2. The 5th term is boxed{dfrac{35 - 15sqrt{5}}{2}}.</think>

🧠:<think>Alright, so I have this problem where a diplomat is negotiating a multi-national treaty, and they want to maximize their country's net benefit function. The function is given as:[ B(x) = sum_{i=1}^{n} a_i x_i - frac{1}{2} sum_{i=1}^{n} sum_{j=1}^{n} b_{ij} x_i x_j ]And there are two parts to the problem. First, I need to determine the conditions on the coefficients ( a_i ) and ( b_{ij} ) such that ( B(x) ) has a unique maximum. Second, I have to find the values of ( x ) that maximize ( B(x) ) under the constraint ( sum_{i=1}^{n} c_i x_i = k ) using Lagrange multipliers.Let me start with the first part.Problem 1: Conditions for a Unique MaximumOkay, so ( B(x) ) is a quadratic function in terms of ( x ). Quadratic functions can have maxima, minima, or saddle points depending on the coefficients. Since the function is quadratic, its graph is a paraboloid, and whether it opens upwards or downwards determines if it has a maximum or a minimum.But wait, in multiple variables, it's a bit more complex. The function is:[ B(x) = mathbf{a}^T mathbf{x} - frac{1}{2} mathbf{x}^T mathbf{B} mathbf{x} ]Where ( mathbf{a} ) is the vector of coefficients ( a_i ), and ( mathbf{B} ) is the matrix with entries ( b_{ij} ).To find the extrema, we can take the gradient of ( B(x) ) and set it to zero. The gradient is:[ nabla B(x) = mathbf{a} - mathbf{B} mathbf{x} ]Setting this equal to zero gives:[ mathbf{B} mathbf{x} = mathbf{a} ]So, the critical point is ( mathbf{x} = mathbf{B}^{-1} mathbf{a} ), provided that ( mathbf{B} ) is invertible.But for this critical point to be a maximum, the function ( B(x) ) must be concave. In quadratic functions, concavity is determined by the matrix ( mathbf{B} ). Specifically, the function is concave if ( mathbf{B} ) is positive definite.Wait, actually, since the quadratic term is subtracted, the function is concave if ( mathbf{B} ) is positive definite. Because the quadratic form ( -frac{1}{2} mathbf{x}^T mathbf{B} mathbf{x} ) will be concave if ( mathbf{B} ) is positive definite.Alternatively, if ( mathbf{B} ) is positive definite, then the Hessian of ( B(x) ) is negative definite (since the Hessian is ( -mathbf{B} )), which implies that the function is concave and hence the critical point is a global maximum.So, the conditions for ( B(x) ) to have a unique maximum are:1. The matrix ( mathbf{B} ) must be positive definite.2. The system ( mathbf{B} mathbf{x} = mathbf{a} ) must have a unique solution, which is guaranteed if ( mathbf{B} ) is invertible, which is already implied by positive definiteness since positive definite matrices are invertible.Therefore, the conditions are that the matrix ( mathbf{B} ) is positive definite.Wait, but let me double-check. The function is quadratic, and the quadratic term is negative, so the function is concave if ( mathbf{B} ) is positive definite. If ( mathbf{B} ) is positive definite, then the function is concave, so any critical point is a global maximum. Since ( mathbf{B} ) is positive definite, it's invertible, so the critical point is unique. Therefore, yes, the function has a unique maximum if and only if ( mathbf{B} ) is positive definite.So, for part 1, the condition is that the matrix ( mathbf{B} ) is positive definite.Problem 2: Maximizing ( B(x) ) under the Constraint ( sum c_i x_i = k )Now, moving on to the second part. We need to maximize ( B(x) ) subject to the linear constraint ( sum_{i=1}^{n} c_i x_i = k ).This is a constrained optimization problem, and the method of Lagrange multipliers is the way to go.First, let's recall how Lagrange multipliers work. We introduce a multiplier ( lambda ) for the constraint and form the Lagrangian:[ mathcal{L}(x, lambda) = B(x) - lambda left( sum_{i=1}^{n} c_i x_i - k right) ]So, substituting ( B(x) ):[ mathcal{L}(x, lambda) = sum_{i=1}^{n} a_i x_i - frac{1}{2} sum_{i=1}^{n} sum_{j=1}^{n} b_{ij} x_i x_j - lambda left( sum_{i=1}^{n} c_i x_i - k right) ]Now, to find the extrema, we take the partial derivatives of ( mathcal{L} ) with respect to each ( x_i ) and ( lambda ), and set them equal to zero.First, let's compute the partial derivative with respect to ( x_i ):[ frac{partial mathcal{L}}{partial x_i} = a_i - sum_{j=1}^{n} b_{ij} x_j - lambda c_i = 0 ]This gives us a system of equations:[ sum_{j=1}^{n} b_{ij} x_j = a_i - lambda c_i quad text{for each } i = 1, 2, ldots, n ]Which can be written in matrix form as:[ mathbf{B} mathbf{x} = mathbf{a} - lambda mathbf{c} ]Where ( mathbf{c} ) is the vector with entries ( c_i ).Additionally, we have the constraint:[ sum_{i=1}^{n} c_i x_i = k ]So, now we have a system of ( n + 1 ) equations:1. ( mathbf{B} mathbf{x} = mathbf{a} - lambda mathbf{c} )2. ( mathbf{c}^T mathbf{x} = k )We can solve this system for ( mathbf{x} ) and ( lambda ).Assuming that ( mathbf{B} ) is invertible (which from part 1, we know it is positive definite, hence invertible), we can express ( mathbf{x} ) as:[ mathbf{x} = mathbf{B}^{-1} (mathbf{a} - lambda mathbf{c}) ]Now, substitute this into the constraint equation:[ mathbf{c}^T mathbf{x} = k ][ mathbf{c}^T mathbf{B}^{-1} (mathbf{a} - lambda mathbf{c}) = k ]Let me denote ( mathbf{c}^T mathbf{B}^{-1} mathbf{a} ) as some scalar, say ( m ), and ( mathbf{c}^T mathbf{B}^{-1} mathbf{c} ) as another scalar, say ( d ).So, expanding the equation:[ mathbf{c}^T mathbf{B}^{-1} mathbf{a} - lambda mathbf{c}^T mathbf{B}^{-1} mathbf{c} = k ][ m - lambda d = k ][ lambda = frac{m - k}{d} ][ lambda = frac{mathbf{c}^T mathbf{B}^{-1} mathbf{a} - k}{mathbf{c}^T mathbf{B}^{-1} mathbf{c}} ]So, now we can substitute ( lambda ) back into the expression for ( mathbf{x} ):[ mathbf{x} = mathbf{B}^{-1} mathbf{a} - lambda mathbf{B}^{-1} mathbf{c} ][ mathbf{x} = mathbf{B}^{-1} mathbf{a} - left( frac{mathbf{c}^T mathbf{B}^{-1} mathbf{a} - k}{mathbf{c}^T mathbf{B}^{-1} mathbf{c}} right) mathbf{B}^{-1} mathbf{c} ]Let me write this more neatly:[ mathbf{x} = mathbf{B}^{-1} mathbf{a} - frac{mathbf{c}^T mathbf{B}^{-1} mathbf{a} - k}{mathbf{c}^T mathbf{B}^{-1} mathbf{c}} mathbf{B}^{-1} mathbf{c} ]Alternatively, we can factor this expression:Let me denote ( mathbf{B}^{-1} mathbf{c} ) as a vector ( mathbf{v} ). Then,[ mathbf{x} = mathbf{B}^{-1} mathbf{a} - frac{mathbf{c}^T mathbf{v} - k}{mathbf{c}^T mathbf{v}} mathbf{v} ]But perhaps it's clearer to leave it in terms of ( mathbf{B}^{-1} ).Alternatively, we can write this as:[ mathbf{x} = mathbf{B}^{-1} mathbf{a} - frac{mathbf{c}^T mathbf{B}^{-1} mathbf{a} - k}{mathbf{c}^T mathbf{B}^{-1} mathbf{c}} mathbf{B}^{-1} mathbf{c} ]This is the expression for ( mathbf{x} ) that maximizes ( B(x) ) under the given constraint.Wait, let me check if this makes sense. If there were no constraint, the maximum would be at ( mathbf{x} = mathbf{B}^{-1} mathbf{a} ). The constraint modifies this by subtracting a term that depends on ( mathbf{c} ) and ( k ). So, this seems reasonable.Alternatively, another way to write this is:[ mathbf{x} = mathbf{B}^{-1} mathbf{a} - mu mathbf{B}^{-1} mathbf{c} ]Where ( mu = frac{mathbf{c}^T mathbf{B}^{-1} mathbf{a} - k}{mathbf{c}^T mathbf{B}^{-1} mathbf{c}} )But perhaps it's better to leave it as is.So, summarizing, the optimal ( mathbf{x} ) is:[ mathbf{x} = mathbf{B}^{-1} mathbf{a} - frac{mathbf{c}^T mathbf{B}^{-1} mathbf{a} - k}{mathbf{c}^T mathbf{B}^{-1} mathbf{c}} mathbf{B}^{-1} mathbf{c} ]Alternatively, we can write this as:[ mathbf{x} = mathbf{B}^{-1} left( mathbf{a} - frac{mathbf{c}^T mathbf{B}^{-1} mathbf{a} - k}{mathbf{c}^T mathbf{B}^{-1} mathbf{c}} mathbf{c} right) ]Which might be a more compact way.Alternatively, factor out ( mathbf{B}^{-1} ):[ mathbf{x} = mathbf{B}^{-1} mathbf{a} - frac{mathbf{c}^T mathbf{B}^{-1} mathbf{a} - k}{mathbf{c}^T mathbf{B}^{-1} mathbf{c}} mathbf{B}^{-1} mathbf{c} ]Yes, that seems correct.Let me verify the dimensions. ( mathbf{B}^{-1} ) is ( n times n ), ( mathbf{a} ) is ( n times 1 ), so ( mathbf{B}^{-1} mathbf{a} ) is ( n times 1 ). Similarly, ( mathbf{B}^{-1} mathbf{c} ) is ( n times 1 ). The scalar ( frac{mathbf{c}^T mathbf{B}^{-1} mathbf{a} - k}{mathbf{c}^T mathbf{B}^{-1} mathbf{c}} ) is just a number, so when multiplied by ( mathbf{B}^{-1} mathbf{c} ), it's still ( n times 1 ). So, the subtraction makes sense, and ( mathbf{x} ) is ( n times 1 ), which is correct.So, I think that's the solution.Double-Checking the Steps1. Formulated the Lagrangian correctly.2. Took partial derivatives with respect to each ( x_i ) and ( lambda ), leading to the system of equations.3. Expressed ( mathbf{x} ) in terms of ( mathbf{B}^{-1} ) and substituted into the constraint to solve for ( lambda ).4. Substituted ( lambda ) back into the expression for ( mathbf{x} ).Everything seems to check out.Alternative Approach Using ProjectionAlternatively, since the constraint is linear, we can think of this as projecting the unconstrained maximum onto the constraint hyperplane. The unconstrained maximum is ( mathbf{x}_0 = mathbf{B}^{-1} mathbf{a} ). The constraint is ( mathbf{c}^T mathbf{x} = k ). The projection can be found by moving along the direction of ( mathbf{c} ) from ( mathbf{x}_0 ) until the constraint is satisfied.The formula for the projection is:[ mathbf{x} = mathbf{x}_0 - frac{mathbf{c}^T mathbf{x}_0 - k}{mathbf{c}^T mathbf{B}^{-1} mathbf{c}} mathbf{B}^{-1} mathbf{c} ]Which is exactly what we derived earlier. So, this alternative perspective confirms our result.ConclusionSo, for part 1, the condition is that the matrix ( mathbf{B} ) is positive definite. For part 2, the optimal ( mathbf{x} ) is given by the expression above.Final Answer1. The net benefit function ( B(x) ) has a unique maximum if and only if the matrix ( mathbf{B} ) is positive definite. Thus, the condition is that ( mathbf{B} ) is positive definite.2. The values of ( x ) that maximize ( B(x) ) under the constraint are:[ boxed{mathbf{x} = mathbf{B}^{-1} mathbf{a} - frac{mathbf{c}^T mathbf{B}^{-1} mathbf{a} - k}{mathbf{c}^T mathbf{B}^{-1} mathbf{c}} mathbf{B}^{-1} mathbf{c}} ]Alternatively, this can be written as:[ boxed{mathbf{x} = mathbf{B}^{-1} left( mathbf{a} - frac{mathbf{c}^T mathbf{B}^{-1} mathbf{a} - k}{mathbf{c}^T mathbf{B}^{-1} mathbf{c}} mathbf{c} right)} ]Either form is acceptable, but the first one is more explicit.Final Answer1. The function ( B(x) ) has a unique maximum if and only if the matrix ( mathbf{B} ) is positive definite. So, the condition is:[ boxed{mathbf{B} text{ is positive definite}} ]2. The optimal values of ( x ) are:[ boxed{mathbf{x} = mathbf{B}^{-1} mathbf{a} - frac{mathbf{c}^T mathbf{B}^{-1} mathbf{a} - k}{mathbf{c}^T mathbf{B}^{-1} mathbf{c}} mathbf{B}^{-1} mathbf{c}} ]</think>

🧠:<think>Okay, so I have this problem about football and basketball formations modeled as directed graphs. Let me try to wrap my head around both parts step by step.Starting with part 1: It's about a football formation with 11 players. Each player can potentially pass to any of the other 10, but in a given play, only a subset of these passes is used. The probability of using any particular pass is independent and equal to p. I need to find the expected number of distinct passing paths from the quarterback to the wide receiver. The play starts with the quarterback and ends with the wide receiver, and a path is a sequence of players where each consecutive pair represents a used pass.Hmm, okay. So, the quarterback is one node, the wide receiver is another, and there are 9 other players in between. Each pass is an edge that's present with probability p, independently. So, the graph is a directed graph with 11 nodes, each node connected to 10 others, but each edge is present with probability p.I need to compute the expected number of paths from quarterback (let's call him Q) to wide receiver (let's call him W). A path is a sequence of players starting at Q and ending at W, with each consecutive pair connected by a used pass.So, expectation is linear, so maybe I can compute the expected number of such paths by considering all possible paths and summing their probabilities.But wait, how many possible paths are there? Since it's a directed graph, each edge is directed, so the passes are directed. So, from Q, you can pass to any of the 10 other players, then from each of those, you can pass to any of the remaining 10, except maybe the previous one? Wait, no, in a directed graph, you can have cycles, so you can pass back to someone you just came from, but in a path, you can revisit nodes? Wait, no, in a path, nodes are distinct? Or can they repeat?Wait, the problem says "a sequence of players starting with the quarterback and ending with the wide receiver, where each consecutive pair of players in the sequence represents a used pass." It doesn't specify whether the players have to be distinct, so I think nodes can be revisited. So, the path can have cycles, but in terms of sequences, it's just any sequence starting at Q and ending at W, with each step being a pass that's used.But wait, in reality, in football, a pass can't go back to the same player, but in the model, it's a directed graph, so edges are directed. So, if Q passes to player A, then player A can pass to Q again, but in the context of a play, that might not make sense, but in the model, it's allowed.But perhaps for the purposes of this problem, we can consider all possible paths, even those with cycles, because the model allows it.But wait, the problem says "distinct passing paths." So, does that mean distinct in terms of the sequence of players, or distinct in terms of the edges used? Hmm, probably distinct in terms of the sequence of players, so even if two paths use different edges but result in the same sequence, they are considered the same path? Or is it about the edges? Hmm, the wording is a bit ambiguous.Wait, the problem says "distinct passing paths from the quarterback to the wide receiver," and a path is a sequence of players starting with Q and ending with W, with each consecutive pair representing a used pass. So, I think a path is defined by the sequence of players, so two paths are different if the sequence of players is different, even if they use the same edges.But in terms of expectation, we can model it as the sum over all possible paths of the probability that all the edges in the path are present.So, the expected number of paths is the sum over all possible paths from Q to W of the product of the probabilities of each edge in the path being present.Since each edge is present with probability p, and edges are independent, the probability that a particular path is present is p^k, where k is the number of edges in the path.So, the expected number of paths is the sum over all possible path lengths k of (number of paths of length k) * p^k.So, let's formalize this.Let me denote Q as node 1 and W as node 11, just for simplicity. So, nodes are 1, 2, ..., 11.We need to compute the expected number of paths from 1 to 11.Each edge is present with probability p, independently.So, the expected number of such paths is the sum over all possible sequences starting at 1, ending at 11, of p^{length of the path}.But how do we compute this sum?This seems similar to computing the expected number of walks from 1 to 11 in a directed graph where each edge is present independently with probability p.In graph theory, the expected number of walks of length k from node i to node j is given by the (i,j) entry of the matrix (A)^k, where A is the adjacency matrix. But since each edge is present with probability p, the expected adjacency matrix is p * J - p * I, where J is the matrix of ones and I is the identity matrix, but actually, since each node can pass to any other node, except itself? Wait, no, in the problem, each player can pass to any of the other 10, so self-loops are not allowed. So, the adjacency matrix has zeros on the diagonal and ones elsewhere, each edge present with probability p.Therefore, the expected number of walks of length k from Q to W is equal to (p * (J - I))^k, evaluated at (Q, W). But since the graph is directed, the adjacency matrix is not symmetric, but in this case, since each node can pass to any other node, it's a complete directed graph without self-loops.Wait, but actually, in a complete directed graph without self-loops, each node has out-degree 10 and in-degree 10.But in our case, each edge is present with probability p, so the adjacency matrix is a random matrix where each off-diagonal entry is 1 with probability p and 0 otherwise.So, the expected number of walks of length k from Q to W is equal to the (Q, W) entry of (p * (J - I))^k.But since the graph is regular, meaning each node has the same number of outgoing and incoming edges, we can perhaps find a closed-form expression.Alternatively, we can model this as a Markov chain and find the expected number of walks.But maybe it's easier to think recursively.Let me denote E_k as the expected number of walks of length k from Q to W.We need to compute the total expected number of walks, which is the sum over k=1 to infinity of E_k.Wait, but in reality, the maximum path length is 10, since there are 11 players, but in theory, the path could cycle indefinitely, but in football, the play would end eventually, but in the model, it's just a graph, so paths can be of any length.But for expectation, we can consider all possible path lengths.But perhaps we can model this as a generating function.Let me think about the adjacency matrix A, where A_{i,j} = 1 if there's an edge from i to j, 0 otherwise, each with probability p.Then, the expected number of walks from Q to W is the sum over k=1 to infinity of (A^k)_{Q,W}.Taking expectation, we have E[sum_{k=1}^infty (A^k)_{Q,W}] = sum_{k=1}^infty E[(A^k)_{Q,W}].But E[A^k] is (p * (J - I))^k, so the expected number of walks is sum_{k=1}^infty (p * (J - I))^k evaluated at (Q, W).But this is a matrix power series, which can be summed up as (I - p * (J - I))^{-1} - I, evaluated at (Q, W).Wait, let me recall that for a matrix A, sum_{k=0}^infty A^k = (I - A)^{-1}, provided that the spectral radius of A is less than 1.So, in our case, sum_{k=1}^infty (p * (J - I))^k = (I - p * (J - I))^{-1} - I.So, the expected number of walks is the (Q, W) entry of (I - p * (J - I))^{-1} - I.But let's compute this.First, let me note that J is the matrix of all ones, and I is the identity matrix.So, p * (J - I) is a matrix where the diagonal entries are 0 and the off-diagonal entries are p.So, the matrix I - p * (J - I) is a matrix where the diagonal entries are 1 and the off-diagonal entries are -p.So, we have to compute the inverse of this matrix.This is a well-known matrix in the context of graph theory and Markov chains. It's a matrix where all diagonal entries are 1 and all off-diagonal entries are -p.Let me denote this matrix as M = I - p * (J - I).So, M = I - p * (J - I) = (1 + p)I - p * J.Wait, no:Wait, I - p*(J - I) = I - pJ + pI = (1 + p)I - pJ.Yes, that's correct.So, M = (1 + p)I - pJ.We need to find M^{-1}.This is a rank-one perturbation of a scaled identity matrix. There is a formula for the inverse of such matrices.Recall that if M = aI + bJ, then M^{-1} = (1/a)I - (b)/(a(a + bn)) J, where n is the size of the matrix.Wait, let me recall the Sherman-Morrison formula or the matrix inversion lemma.Alternatively, for a matrix of the form M = cI + dJ, the inverse can be computed as follows.Let me denote n = 11, the number of players.So, M = (1 + p)I - pJ.We can write M = (1 + p)I - pJ.Let me factor out (1 + p):M = (1 + p)(I - (p/(1 + p)) J).So, M^{-1} = (1/(1 + p)) (I - (p/(1 + p)) J)^{-1}.Now, we need to compute (I - (p/(1 + p)) J)^{-1}.Let me denote q = p/(1 + p).So, we have (I - q J)^{-1}.The inverse of I - q J can be computed as follows.Note that J is a rank-one matrix, so we can use the Sherman-Morrison formula.The Sherman-Morrison formula states that (I + uv^T)^{-1} = I - uv^T / (1 + v^T u).In our case, I - q J = I - q * (ee^T), where e is the column vector of ones.So, u = -q e, v^T = e^T.Thus, (I - q J)^{-1} = I + (q e e^T) / (1 + q e^T e).Since e^T e = n = 11.So, (I - q J)^{-1} = I + (q e e^T) / (1 + 11 q).Therefore, M^{-1} = (1/(1 + p)) [I + (q e e^T)/(1 + 11 q)].Substituting back q = p/(1 + p):M^{-1} = (1/(1 + p)) [I + ( (p/(1 + p)) e e^T ) / (1 + 11*(p/(1 + p)) ) ].Simplify the denominator:1 + 11*(p/(1 + p)) = (1 + p + 11 p)/(1 + p) = (1 + 12 p)/(1 + p).So, M^{-1} = (1/(1 + p)) [I + ( (p/(1 + p)) e e^T ) * ( (1 + p)/(1 + 12 p) ) ) ].Simplify:M^{-1} = (1/(1 + p)) [I + (p e e^T)/(1 + 12 p) ) ].So, M^{-1} = (1/(1 + p)) I + (p/( (1 + p)(1 + 12 p) )) e e^T.Now, we need the (Q, W) entry of M^{-1} - I, because the expected number of walks is sum_{k=1}^infty (p * (J - I))^k, which is M^{-1} - I.Wait, no, earlier we had sum_{k=1}^infty (p * (J - I))^k = M^{-1} - I, where M = I - p*(J - I).So, the expected number of walks is (M^{-1} - I)_{Q,W}.So, let's compute M^{-1} - I.From above, M^{-1} = (1/(1 + p)) I + (p/( (1 + p)(1 + 12 p) )) e e^T.Therefore, M^{-1} - I = (1/(1 + p) - 1) I + (p/( (1 + p)(1 + 12 p) )) e e^T.Simplify:1/(1 + p) - 1 = (1 - (1 + p))/(1 + p) = (-p)/(1 + p).So, M^{-1} - I = (-p/(1 + p)) I + (p/( (1 + p)(1 + 12 p) )) e e^T.Now, we need the (Q, W) entry of this matrix.Since Q and W are two distinct nodes, the (Q, W) entry is:(-p/(1 + p)) * delta_{Q,W} + (p/( (1 + p)(1 + 12 p) )) * (e e^T)_{Q,W}.But since Q ≠ W, delta_{Q,W} = 0, and (e e^T)_{Q,W} = 1, because e is a vector of ones.Therefore, the (Q, W) entry is:0 + (p/( (1 + p)(1 + 12 p) )) * 1 = p / [ (1 + p)(1 + 12 p) ].So, the expected number of walks from Q to W is p / [ (1 + p)(1 + 12 p) ].Wait, that seems too simple. Let me verify.Wait, actually, in the expression M^{-1} - I, the (Q, W) entry is p / [ (1 + p)(1 + 12 p) ].But let me think about the case when p is very small. If p approaches 0, the expected number of paths should approach the number of direct passes from Q to W, which is 1 edge, so expectation p. But according to our formula, when p approaches 0, p / [ (1 + p)(1 + 12 p) ] ≈ p / (1 * 1) = p, which matches. So that seems correct.Another check: when p = 1/12, the denominator becomes (1 + 1/12)(1 + 1) = (13/12)(2) = 13/6, so the expectation is (1/12) / (13/6) = (1/12)*(6/13) = 1/26. Hmm, not sure if that makes sense, but let's see.Alternatively, maybe I made a mistake in the calculation.Wait, let's go back.We had M = I - p*(J - I) = (1 + p)I - pJ.Then, M^{-1} = (1/(1 + p)) [I + (p e e^T)/(1 + 11 p/(1 + p)) ].Wait, let me re-express the denominator correctly.Earlier, I had:1 + 11 q = 1 + 11*(p/(1 + p)) = (1 + p + 11 p)/(1 + p) = (1 + 12 p)/(1 + p).So, that part is correct.Then, M^{-1} = (1/(1 + p)) [I + (p/(1 + p)) e e^T * (1 + p)/(1 + 12 p) ) ].Which simplifies to (1/(1 + p)) I + (p/(1 + 12 p)) e e^T.Wait, no, let me re-express:M^{-1} = (1/(1 + p)) [I + (p/(1 + p)) e e^T / (1 + 11*(p/(1 + p)) ) ].Which is (1/(1 + p)) [I + (p/(1 + p)) e e^T / ( (1 + 12 p)/(1 + p) ) ) ].So, the denominator is (1 + 12 p)/(1 + p), so dividing by that is multiplying by (1 + p)/(1 + 12 p).Thus, M^{-1} = (1/(1 + p)) [I + (p/(1 + p)) e e^T * (1 + p)/(1 + 12 p) ) ].Simplify:The (1 + p) cancels out in the numerator and denominator:M^{-1} = (1/(1 + p)) [I + (p e e^T)/(1 + 12 p) ) ].So, yes, that's correct.Then, M^{-1} - I = (1/(1 + p) - 1) I + (p/( (1 + p)(1 + 12 p) )) e e^T.Which is (-p/(1 + p)) I + (p/( (1 + p)(1 + 12 p) )) e e^T.So, for the (Q, W) entry, since Q ≠ W, it's the second term: p / [ (1 + p)(1 + 12 p) ].Therefore, the expected number of paths from Q to W is p / [ (1 + p)(1 + 12 p) ].Wait, but this seems counterintuitive because when p is 1, meaning all edges are present, the expected number of paths should be infinite, but our formula gives 1 / (2 * 13) = 1/26, which is finite. That can't be right.Wait, that must mean I made a mistake in interpreting the problem.Wait, no, when p = 1, the graph is complete, so there are infinitely many paths from Q to W because you can have cycles. But in reality, in a complete directed graph without self-loops, the number of paths of length k from Q to W is 10^{k-1}, because from Q, you can go to any of the 10 players, then from there to any of the 10, etc., until the last step goes to W.But since the graph is complete, the number of paths is indeed infinite when considering all possible path lengths. However, in our formula, when p = 1, we get 1/(2 * 13) = 1/26, which is finite. That suggests that our approach is flawed.Wait, perhaps the issue is that when p = 1, the matrix M = I - p*(J - I) becomes I - (J - I) = 2I - J. Then, M^{-1} would be the inverse of 2I - J, but in reality, when p = 1, the expected number of walks is indeed infinite because the graph is strongly connected and has cycles, leading to infinitely many paths. So, our formula must be incorrect.Wait, perhaps the problem is that we're considering the expected number of walks, but when p is such that the graph has a directed cycle, the number of walks becomes infinite. However, in our case, since we're dealing with a finite graph, but with possible cycles, the expected number of walks is actually finite only if the spectral radius of the adjacency matrix is less than 1. But in our case, the adjacency matrix has a spectral radius that depends on p.Wait, but in our calculation, we assumed that the series converges, which it does if the spectral radius of p*(J - I) is less than 1. The spectral radius of J - I is 10, because J is a rank-one matrix with eigenvalues 11 (with multiplicity 1) and 0 (with multiplicity 10). So, J - I has eigenvalues 10 (with multiplicity 1) and -1 (with multiplicity 10). Therefore, the spectral radius of p*(J - I) is 10p. So, for convergence, we need 10p < 1, i.e., p < 1/10.So, our formula is valid only when p < 1/10. When p >= 1/10, the series doesn't converge, and the expected number of walks is infinite.But in the problem statement, p is just given as a probability, so it's between 0 and 1. So, perhaps the answer is conditional on p < 1/10.Alternatively, maybe the problem expects a different approach, considering only simple paths, i.e., paths without cycles, which would make the number of paths finite.Wait, the problem says "distinct passing paths," but it doesn't specify whether they are simple paths or not. So, perhaps the intended answer is for simple paths, i.e., paths without repeating nodes.In that case, the number of simple paths from Q to W is finite, and we can compute the expected number by summing over all possible simple paths.But that complicates things because the number of simple paths grows factorially with the number of nodes, and computing the expectation would require inclusion-exclusion or something similar.Alternatively, maybe the problem is considering only paths of a certain length, but it's not specified.Wait, let me re-read the problem."Derive the expected number of distinct passing paths from the quarterback to the wide receiver, assuming the play starts with the ball at the quarterback and ends with a catch by the wide receiver. A path is a sequence of players starting with the quarterback and ending with the wide receiver, where each consecutive pair of players in the sequence represents a used pass."So, it doesn't specify that the path must be simple, so cycles are allowed. Therefore, the number of paths is indeed infinite when p >= 1/10, but the expectation can be expressed as a finite value when p < 1/10.But in our earlier calculation, we got p / [ (1 + p)(1 + 12 p) ].Wait, let me test p = 0. So, when p = 0, the expected number of paths is 0, which is correct because no passes are used.When p approaches 1/10 from below, the denominator approaches (1 + 1/10)(1 + 12*(1/10)) = (11/10)(22/10) = (11/10)(11/5) = 121/50 ≈ 2.42, so the expectation approaches (1/10) / (121/50) = (1/10)*(50/121) = 5/121 ≈ 0.0413.But when p = 1/10, the spectral radius is exactly 1, so the series doesn't converge, and the expectation is infinite.So, perhaps the answer is p / [ (1 + p)(1 + 12 p) ] for p < 1/10, and infinite otherwise.But the problem doesn't specify any constraints on p, so maybe we can just write the formula as p / [ (1 + p)(1 + 12 p) ].Alternatively, perhaps I made a mistake in the calculation.Wait, let me think differently.Let me model this as a branching process. Starting from Q, each step, a player can pass to any of the other 10 players with probability p. So, the expected number of paths from Q to W can be modeled recursively.Let me denote E as the expected number of paths from Q to W.From Q, the ball can go to any of the other 10 players with probability p each. So, the expected number of paths is the sum over all possible next players of the probability of passing to them times the expected number of paths from there to W.But wait, that's similar to E = sum_{i ≠ Q} p * (1 + E_i), where E_i is the expected number of paths from player i to W.But actually, from Q, the ball can go to any of the 10 players, and from each of those, the ball can go to any of the 10 players, including back to Q, and so on, until it reaches W.So, let me define E as the expected number of paths from Q to W.Then, E = p * (number of players other than Q) * (1 + E'), where E' is the expected number of paths from a non-Q, non-W player to W.Wait, but W is a specific player, so if we reach W, the path ends.Wait, perhaps it's better to define E as the expected number of paths from Q to W, and F as the expected number of paths from a non-Q, non-W player to W.So, starting from Q, the ball can go to any of the 10 players (including W) with probability p each.If it goes to W, that's a path of length 1, contributing 1.If it goes to another player (not Q or W), then from there, the expected number of paths is F.So, E = p * 1 + p * 9 * (1 + F).Wait, no, because from Q, you can go to W directly, contributing 1 path, or go to one of the other 9 players, and from each of those, you can have paths that go through them.But actually, the expected number of paths is the sum over all possible paths, so it's not just the expected number of paths of length 1, but all lengths.Wait, perhaps a better way is to set up equations for E and F.Let me define:E: expected number of paths from Q to W.F: expected number of paths from a non-Q, non-W player to W.So, from Q, the ball can go to W with probability p, contributing 1 path.From Q, the ball can go to any of the other 9 players (non-Q, non-W) with probability p each, and from each of those, the expected number of paths is F.Additionally, from Q, the ball can go back to Q, but since we're considering paths starting at Q, going back to Q would create a cycle, but in terms of paths, it's allowed.Wait, but in our initial definition, E is the expected number of paths from Q to W, regardless of the path length. So, if the ball goes back to Q, it can continue to pass around, potentially forever.Therefore, the equation for E is:E = p * 1 + p * 9 * F + p * 1 * E.Wait, because from Q, with probability p, you can pass to W, contributing 1 path.With probability p, you can pass to each of the 9 other non-W players, contributing F each.With probability p, you can pass back to Q, contributing E again.Wait, but there are 10 possible passes from Q: to W, to each of the 9 others, and to Q itself? Wait, no, in the problem, each player can pass to any of the other 10 players, so self-loops are allowed? Wait, no, the problem says "each player can potentially pass to any of the other 10 players," so self-loops are not allowed. So, from Q, you can pass to any of the other 10 players, which includes W and the 9 others, but not Q.Therefore, the equation is:E = p * 1 + p * 9 * F.Because from Q, you can pass to W with probability p, contributing 1 path, or pass to one of the 9 other players with probability p each, contributing F each.Similarly, from a non-Q, non-W player (let's say player X), the expected number of paths to W is F.From X, the ball can go to W with probability p, contributing 1 path.From X, the ball can go to Q with probability p, contributing E.From X, the ball can go to any of the other 8 players (non-Q, non-W) with probability p each, contributing F each.So, the equation for F is:F = p * 1 + p * 1 * E + p * 8 * F.So, we have two equations:1. E = p + 9 p F2. F = p + p E + 8 p FLet me write them as:E = p + 9 p F ...(1)F = p + p E + 8 p F ...(2)Let me solve equation (2) for F.F = p + p E + 8 p FSubtract 8 p F from both sides:F - 8 p F = p + p EFactor F:F (1 - 8 p) = p (1 + E)So,F = [ p (1 + E) ] / (1 - 8 p) ...(2a)Now, substitute F from (2a) into equation (1):E = p + 9 p * [ p (1 + E) / (1 - 8 p) ]Simplify:E = p + (9 p^2 (1 + E)) / (1 - 8 p)Multiply both sides by (1 - 8 p):E (1 - 8 p) = p (1 - 8 p) + 9 p^2 (1 + E)Expand:E - 8 p E = p - 8 p^2 + 9 p^2 + 9 p^2 ESimplify the right-hand side:p - 8 p^2 + 9 p^2 = p + p^2So,E - 8 p E = p + p^2 + 9 p^2 EBring all terms to the left:E - 8 p E - p - p^2 - 9 p^2 E = 0Factor E:E (1 - 8 p - 9 p^2) - p - p^2 = 0So,E (1 - 8 p - 9 p^2) = p + p^2Therefore,E = (p + p^2) / (1 - 8 p - 9 p^2)Factor numerator and denominator:Numerator: p (1 + p)Denominator: 1 - 8 p - 9 p^2Let me factor the denominator:1 - 8 p - 9 p^2 = -(9 p^2 + 8 p - 1) = -(9 p^2 + 9 p - p -1) = -(9 p (p + 1) -1 (p +1)) = -(9 p -1)(p +1)Wait, let me check:Let me write 9 p^2 + 8 p -1.Looking for factors of 9*(-1) = -9 that add up to 8.Factors: 9 and -1.So, 9 p^2 + 9 p - p -1 = 9 p (p +1) -1 (p +1) = (9 p -1)(p +1)Therefore,1 - 8 p - 9 p^2 = -(9 p^2 +8 p -1) = -(9 p -1)(p +1)So,E = p (1 + p) / [ - (9 p -1)(p +1) ) ] = - p (1 + p) / [ (9 p -1)(p +1) ) ] = - p / (9 p -1 )Simplify:E = p / (1 - 9 p )So, the expected number of paths is E = p / (1 - 9 p )Wait, that's different from our earlier result. Which one is correct?Let me test p = 0. Then E = 0, which is correct.When p approaches 1/9 from below, E approaches infinity, which makes sense because at p = 1/9, the denominator becomes zero, indicating a phase transition.Wait, but earlier, using the matrix approach, we had E = p / [ (1 + p)(1 + 12 p) ].But now, using the recursive approach, we have E = p / (1 - 9 p).Which one is correct?Let me test p = 1/10.Using the recursive approach: E = (1/10) / (1 - 9/10) = (1/10)/(1/10) = 1.Using the matrix approach: E = (1/10) / [ (1 + 1/10)(1 + 12*(1/10)) ] = (1/10) / [ (11/10)(22/10) ] = (1/10) / (242/100) = (1/10)*(100/242) = 10/242 ≈ 0.0413.But when p = 1/10, the expected number of paths should be finite, but according to the recursive approach, it's 1, which seems low.Wait, but in reality, when p = 1/10, the expected number of paths is the sum over k=1 to infinity of (number of paths of length k) * (1/10)^k.The number of paths of length 1: 1 (direct pass).Number of paths of length 2: 9 (from Q to any of the 9 non-W players, then to W).Number of paths of length 3: 9*9 (from Q to any of 9, then to any of 9, then to W).Wait, no, because from the second player, you can go back to Q or to any of the other 9, but in the recursive approach, we considered F as the expected number from a non-Q, non-W player.Wait, in the recursive approach, we found E = p / (1 - 9 p).So, for p = 1/10, E = (1/10)/(1 - 9/10) = 1.But let's compute the actual expected number:E = sum_{k=1}^infty (number of paths of length k) * p^k.Number of paths of length 1: 1.Number of paths of length 2: 9.Number of paths of length 3: 9*9 = 81.Number of paths of length 4: 9^3 = 729.And so on.So, E = 1*p + 9*p^2 + 81*p^3 + 729*p^4 + ... = sum_{k=1}^infty 9^{k-1} p^k.This is a geometric series with first term p and common ratio 9 p.So, E = p / (1 - 9 p), which matches the recursive approach.Therefore, the recursive approach is correct, and the matrix approach must have been flawed.Wait, why did the matrix approach give a different result?In the matrix approach, we considered the expected number of walks, but perhaps we didn't account for the fact that once we reach W, the path ends. So, in the matrix approach, we considered all walks from Q to W, including those that pass through W multiple times, but in reality, once the ball reaches W, the play ends, so paths cannot go beyond W.Therefore, the matrix approach incorrectly allows paths that go through W multiple times, whereas in reality, once W is reached, the path terminates.Therefore, the recursive approach is more accurate because it models the termination at W.So, the correct expected number of paths is E = p / (1 - 9 p).Therefore, the answer to part 1 is E = p / (1 - 9 p).But let me verify this with p = 1/10:E = (1/10)/(1 - 9/10) = (1/10)/(1/10) = 1.Which makes sense because:E = sum_{k=1}^infty 9^{k-1}*(1/10)^k = (1/10) sum_{k=0}^infty (9/10)^k = (1/10)*(1/(1 - 9/10)) = (1/10)*(10) = 1.Yes, that's correct.So, the matrix approach was incorrect because it didn't account for the termination at W, whereas the recursive approach correctly models the process where reaching W stops the path.Therefore, the expected number of distinct passing paths from Q to W is E = p / (1 - 9 p).Now, moving on to part 2."For a basketball play involving 5 players, the coach assumes each player can pass to any of the other 4 players, but the average basketball play utilizes only about 3 passes. For a given basketball play, calculate the probability that a randomly selected sequence of 3 passes forms a valid path from the point guard to the shooting guard, assuming equal likelihood for each sequence of passes. Consider that the total possible passing sequences in basketball are fewer than in football due to the smaller number of players and passes."So, we have 5 players, each can pass to any of the other 4. A play consists of 3 passes, and we need to find the probability that a randomly selected sequence of 3 passes forms a valid path from point guard (PG) to shooting guard (SG).Assuming equal likelihood for each sequence, so each sequence of 3 passes is equally likely.First, let's model this as a directed graph with 5 nodes, each node connected to the other 4, so each node has out-degree 4.A sequence of 3 passes is a sequence of 4 players: PG -> A -> B -> SG, where A and B are any players (could be the same or different, could include PG or SG in between).But wait, a sequence of 3 passes involves 4 players, starting at PG and ending at SG.But in the problem, it's a sequence of 3 passes, so it's a path of length 3, meaning 4 nodes.But the problem says "a randomly selected sequence of 3 passes forms a valid path from PG to SG."So, we need to count the number of valid sequences of 3 passes that start at PG and end at SG, divided by the total number of possible sequences of 3 passes.But first, let's clarify:Each pass is a directed edge from one player to another. Since each player can pass to any of the other 4, each pass has 4 possibilities.A sequence of 3 passes is a sequence of 3 directed edges, where each edge starts where the previous one ended.But in the problem, it's a "sequence of passes," so it's a path where each consecutive pass starts where the previous one ended.But the problem says "a randomly selected sequence of 3 passes forms a valid path from PG to SG."So, the total number of possible sequences of 3 passes is 4^3 = 64, because each pass has 4 choices.But wait, no, because each pass is dependent on the previous one. The first pass can be any of the 4 players (since PG can pass to any of the other 4). The second pass is from the receiver of the first pass, which can pass to any of the other 4, including PG or the first receiver. The third pass is similarly from the receiver of the second pass.But in terms of sequences, the total number of possible sequences of 3 passes is 4 * 4 * 4 = 64, because each pass is independent in terms of choice, but dependent in terms of starting point.Wait, no, actually, the number of sequences is 4 * 4 * 4 = 64, because for each pass, you have 4 choices, regardless of where you are.But actually, no, because the starting point of each pass depends on the previous pass.Wait, let me think carefully.The first pass is from PG, so there are 4 choices.The second pass is from the receiver of the first pass, who can pass to any of the other 4 players, so 4 choices.The third pass is from the receiver of the second pass, again 4 choices.Therefore, the total number of sequences is 4 * 4 * 4 = 64.But the number of valid sequences that start at PG and end at SG is the number of paths of length 3 from PG to SG.So, we need to count the number of such paths.Let me denote the players as PG, SG, and three others: A, B, C.So, we have 5 players: PG, A, B, C, SG.We need to count the number of paths of length 3 from PG to SG.A path of length 3 is a sequence of 4 players: PG -> X -> Y -> SG.Where X and Y can be any of the 5 players, including PG and SG, except that you can't pass to yourself.Wait, no, in the problem, each player can pass to any of the other 4, so self-passes are not allowed.Therefore, in the sequence PG -> X -> Y -> SG, X can be A, B, C, or SG, but not PG.Similarly, Y can be any of the other 4 players except the one they received from.Wait, no, the only restriction is that each pass must be to a different player, so in the sequence, you can have repeats except for consecutive repeats.Wait, no, in a pass, you can pass to any of the other 4, so you can pass back to the previous player.So, for example, PG -> A -> PG -> SG is a valid sequence.So, the number of paths of length 3 from PG to SG is equal to the number of walks of length 3 from PG to SG in the complete directed graph of 5 nodes without self-loops.In such a graph, each node has out-degree 4.The number of walks of length 3 from PG to SG can be computed using matrix multiplication.Let me denote the adjacency matrix A, where A_{i,j} = 1 if there is an edge from i to j, 0 otherwise.Since it's a complete directed graph without self-loops, A is a 5x5 matrix with 0s on the diagonal and 1s elsewhere.Then, the number of walks of length 3 from PG to SG is (A^3)_{PG, SG}.But since the graph is regular, we can compute this using the properties of the adjacency matrix.Alternatively, we can compute it recursively.Let me denote:Let a_n be the number of walks of length n from PG to SG.Let b_n be the number of walks of length n from PG to any other non-SG player.So, starting from PG:- For n=0, a_0 = 0 (since we're at PG, not SG), and b_0 = 1 (since we're at PG, which is a non-SG player? Wait, no.Wait, actually, at n=0, we're at PG, so a_0 = 0 (not at SG), and the number of walks of length 0 from PG to non-SG players is 0, except for PG itself.Wait, perhaps a better approach is to define:Let a_n be the number of walks of length n ending at SG.Let b_n be the number of walks of length n ending at non-SG players.So, starting from PG:At n=0:a_0 = 0 (since we're at PG, not SG).b_0 = 1 (since we're at PG, which is a non-SG player).At n=1:From PG, we can go to any of the 4 other players, none of which is SG (since we're at n=1, the first pass).Wait, no, PG can pass to SG in the first pass.Wait, no, in the first pass, PG can pass to any of the 4 other players, which includes SG.So, at n=1:a_1 = 1 (PG -> SG).b_1 = 3 (PG -> A, PG -> B, PG -> C).Wait, no, PG can pass to 4 players: A, B, C, SG.So, a_1 = 1 (PG -> SG).b_1 = 3 (PG -> A, PG -> B, PG -> C).At n=2:From SG, you can pass to any of the other 4 players, none of which is SG (since you can't pass to yourself).So, from SG, you can pass to PG, A, B, C.Similarly, from A, B, C, you can pass to any of the other 4 players, including SG.So, the recurrence relations are:a_{n} = number of walks of length n ending at SG.To get to SG at step n, you must have been at some non-SG player at step n-1, and then passed to SG.Since from any non-SG player, you can pass to SG with 1 possibility.Therefore, a_n = b_{n-1} * 1.Similarly, b_n is the number of walks of length n ending at non-SG players.To get to a non-SG player at step n, you can come from SG or from another non-SG player.From SG, you can pass to 4 non-SG players.From each non-SG player, you can pass to 3 other non-SG players (since you can't pass to yourself).Wait, no, from a non-SG player, you can pass to any of the other 4 players, which includes SG and the other 3 non-SG players.So, from a non-SG player, the number of ways to end at a non-SG player is 3 (since you can't pass to yourself, so 4 -1 = 3).Therefore, the recurrence relations are:a_n = b_{n-1} * 1b_n = a_{n-1} * 4 + b_{n-1} * 3Because:- From SG (a_{n-1}), you can pass to 4 non-SG players.- From non-SG players (b_{n-1}), you can pass to 3 other non-SG players.So, we have:a_n = b_{n-1}b_n = 4 a_{n-1} + 3 b_{n-1}We can use these to compute a_3.Given:At n=0:a_0 = 0b_0 = 1At n=1:a_1 = b_0 = 1b_1 = 4 a_0 + 3 b_0 = 0 + 3*1 = 3At n=2:a_2 = b_1 = 3b_2 = 4 a_1 + 3 b_1 = 4*1 + 3*3 = 4 + 9 = 13At n=3:a_3 = b_2 = 13b_3 = 4 a_2 + 3 b_2 = 4*3 + 3*13 = 12 + 39 = 51Therefore, the number of walks of length 3 from PG to SG is a_3 = 13.But wait, let me verify this.Alternatively, we can compute the number of walks using the adjacency matrix.The adjacency matrix A is a 5x5 matrix with 0s on the diagonal and 1s elsewhere.The number of walks of length 3 from PG to SG is (A^3)_{PG, SG}.But computing A^3 is a bit tedious, but let's try.The number of walks of length 3 from PG to SG is equal to the sum over all possible intermediate players X and Y of the number of walks PG -> X -> Y -> SG.Each step, the number of choices depends on the current player.But perhaps it's easier to use the recurrence relations.From above, we have a_3 = 13.Therefore, the number of valid sequences is 13.The total number of possible sequences is 4^3 = 64.Therefore, the probability is 13/64.But let me verify this.Wait, another way to compute the number of walks is to note that in a complete directed graph of 5 nodes without self-loops, the number of walks of length 3 from PG to SG is equal to:From PG, first pass: 4 choices (including SG).If first pass is to SG, then the next two passes must go from SG to somewhere and then back to SG, but wait, no, because we're counting walks of length 3, so it's PG -> X -> Y -> SG.So, the number of such walks is:Case 1: X ≠ SG.Then, PG -> X (3 choices: A, B, C).From X, you can go to any of the 4 players, including SG.If from X, you go to SG, then the third pass must go from SG to somewhere, but we need to end at SG, so the third pass must be from SG to SG, which is not allowed (self-pass). Therefore, if X ≠ SG and Y = SG, then the third pass cannot be made, so such walks are invalid.Wait, no, in the walk PG -> X -> Y -> SG, Y can be anything, but the third pass must go to SG.So, if X ≠ SG, then from X, you can go to SG or to another player.If from X, you go to SG, then the third pass is from SG to SG, which is invalid.If from X, you go to another player Z, then from Z, you can go to SG.So, let's break it down.Case 1: X ≠ SG (i.e., X is A, B, or C).From PG, choose X: 3 choices.From X, choose Y:- If Y = SG: Then, from SG, you can't pass to SG, so this path is invalid.- If Y ≠ SG: Then, from Y, you can pass to SG or not.Wait, no, the walk is PG -> X -> Y -> SG.So, Y can be any player, but the third pass must be from Y to SG.So, if Y = SG, then the third pass is from SG to SG, which is invalid.If Y ≠ SG, then the third pass is from Y to SG, which is valid.Therefore, for each X ≠ SG:- From X, Y can be any of the 4 players except X.- If Y = SG: invalid.- If Y ≠ SG: valid.So, for each X ≠ SG:Number of valid Y: 3 (since Y can be PG, A, B, C, excluding X and SG).Wait, no, Y can be any of the 4 players except X.But Y must not be X, but can be SG or not.Wait, no, Y can be any of the 4 players except X.So, for each X ≠ SG:- Y can be SG or any of the other 3 non-X players.But if Y = SG, then the third pass is invalid.If Y ≠ SG, then the third pass is valid.Therefore, for each X ≠ SG:Number of valid Y: 3 (since Y can be PG, A, B, C, excluding X and SG).Wait, no, Y can be any of the 4 players except X, so:- If X ≠ SG, then Y can be SG or any of the other 3 non-X players.But if Y = SG, the third pass is invalid.If Y ≠ SG, the third pass is valid.Therefore, for each X ≠ SG:Number of valid Y: 3 (since Y can be PG, A, B, C, excluding X and SG).Wait, no, Y can be any of the 4 players except X, so:- If X ≠ SG, then Y can be SG or any of the other 3 non-X players.But if Y = SG, the third pass is invalid.If Y ≠ SG, the third pass is valid.Therefore, for each X ≠ SG:Number of valid Y: 3 (since Y can be PG, A, B, C, excluding X and SG).Wait, no, Y can be any of the 4 players except X, so:- If X ≠ SG, then Y can be SG or any of the other 3 non-X players.But if Y = SG, the third pass is invalid.If Y ≠ SG, the third pass is valid.Therefore, for each X ≠ SG:Number of valid Y: 3 (since Y can be PG, A, B, C, excluding X and SG).Wait, but Y can be SG, which is invalid, so the number of valid Y is 3.Therefore, for each X ≠ SG:Number of valid walks: 3.Since X has 3 choices (A, B, C), total for Case 1: 3 * 3 = 9.Case 2: X = SG.From PG, choose X = SG: 1 choice.From SG, choose Y:- Y can be any of the 4 players except SG.So, Y can be PG, A, B, C.From Y, we need to go to SG.So, for each Y ≠ SG:From Y, you can pass to SG.Therefore, for each Y ≠ SG, there is 1 valid pass.Since Y can be PG, A, B, C, that's 4 choices.Therefore, for Case 2:Number of valid walks: 1 (X=SG) * 4 (Y) * 1 (Z=SG) = 4.But wait, in this case, the walk is PG -> SG -> Y -> SG.But from SG, you can pass to Y, and then from Y, you can pass to SG.So, for each Y ≠ SG, there is 1 walk.Therefore, total for Case 2: 4.Therefore, total number of valid walks: Case 1 + Case 2 = 9 + 4 = 13.Which matches our earlier result.Therefore, the number of valid sequences is 13.The total number of possible sequences is 4^3 = 64.Therefore, the probability is 13/64.So, the answer to part 2 is 13/64.</think>