1994 AHSME Problems/Problem 22

Problem

Nine chairs in a row are to be occupied by six students and Professors Alpha, Beta and Gamma. These three professors arrive before the six students and decide to choose their chairs so that each professor will be between two students. In how many ways can Professors Alpha, Beta and Gamma choose their chairs?

$\textbf{(A)}\ 12 \qquad\textbf{(B)}\ 36 \qquad\textbf{(C)}\ 60 \qquad\textbf{(D)}\ 84 \qquad\textbf{(E)}\ 630$

Solution

Since each professor must sit between two students, they cannot be seated in seats $1$ or $9$ (the seats at either end of the row). Hence, each professor has $7$ seats they can choose from and must be at least $1$ seat apart.

This question is equivalent to choosing $3$ seats from a row of $5$ seats with no restrictions because we can simply generate a valid arrangement by inserting a seat right after the first and second seat chosen.

Hence, the answer is $\binom{5}{3} \cdot 3! =$ $60$ $\textbf{(C)}$.

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