2009 AMC 12B Problems/Problem 21

Problem

Ten women sit in $10$ seats in a line. All of the $10$ get up and then reseat themselves using all $10$ seats, each sitting in the seat she was in before or a seat next to the one she occupied before. In how many ways can the women be reseated?

$\textbf{(A)}\ 89\qquad \textbf{(B)}\ 90\qquad \textbf{(C)}\ 120\qquad \textbf{(D)}\ 2^{10}\qquad \textbf{(E)}\ 2^2 3^8$

Solution

Let $W_n$ be the answer for $n$ women, we want to find $W_{10}$.

Clearly $W_0=W_1=1$. Now let $n>1$. Let the row of seats go from left to right. Label both the seats and the women $1$ to $n$, going from left to right. Consider the rightmost seat. Which women can sit there after the swap? It can either be woman $n$ or woman $n-1$, as for any other woman the seat is too far.

If woman $n$ stays in her seat, there are exactly $W_{n-1}$ valid arrangements of the other $n-1$ women. If woman $n-1$ sits on seat $n$, we only have one option for woman $n$: she must take seat $n-1$, all the other seats are too far for her. We are left with women $1$ to $n-2$ sitting on seats $1$ to $n-2$, and there are clearly $W_{n-2}$ valid arrangements of these.

We get the recurrence $W_n = W_{n-1} + W_{n-2}$. (Hence $W_n$ is precisely the $n$-th Fibonacci number.) Using this recurrence we can easily compute that $W_{10} = \boxed{89}$.

See Also

2009 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 20
Followed by
Problem 22
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All AMC 12 Problems and Solutions

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