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Find the number of ways to distribute $n$ distinct objects to $5$ distinct boxes, such that boxes $1$, $3$ and $5$ must hold an odd number of objects and boxes $2$ and $4$ must hold an even number of objects.

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Is the value of $n$ odd or even?

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If $n$ is odd, this question is quite hard. One approach is as follows.

Let $A_{n,i}$ be the number of ways of distributing $n$ objects between 5 boxes such that $i$ of the boxes contain an odd number of objects ($0 \leq i \leq 5$). Then

$$\vec{A_n} = \begin{pmatrix} 0&1&0&0&0&0\\ 5&0&2&0&0&0\\ 0&4&0&3&0&0\\ 0&0&3&0&4&0\\ 0&0&0&2&0&5\\ 0&0&0&0&1&0 \end{pmatrix}^n \begin{pmatrix}1\\0\\0\\0\\0\\0\end{pmatrix}$$

We want $A_{n,3} / {5 \choose 3}$.

One way to compute the power of the matrix is to diagonalise it. It's an automatic procedure, but for a $6×6$ matrix I would not want to do it by hand!

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