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?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The two end chairs must be occupied by students, so the professors have seven middle chairs from which to choose, with no two adjacent. If these chairs are numbered from to , the three chairs can be:
Within each triple, the professors can arrange themselves in ways, so the total number is .
Imagine the six students standing in a row before they are seated. There are spaces between them, each of which may be occupied by at most one of the professors. Therefore, there are ways the three professors can select their places.