1994 AMC 12 Problem 22

Question

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?

12

36

60

84

630

Answer: C

Concepts:seating arrangements nonconsecutive selections

Difficulty rating: 1900

Small hint:

Professor chairs cannot be endpoints or adjacent to one another

Big hint:

First choose three nonconsecutive positions from chairs

2

through

8

, then assign the professors

Solution:

The professors must occupy three nonconsecutive positions among chairs

2,3,\ldots,8.

The number of such

3

-subsets of

7

consecutive positions is

\binom{7-3+1}{3}=\binom53=10.

The three distinct professors can be assigned to the selected chairs in

3!=6

ways, for

10\cdot6=60.

Thus the correct answer is C.