Solution: 2015 Fall Midterm - 1

Author: Michiel Smid

Question

The Carleton Computer Science Society has a Board of Directors consisting of a President, two Vice-Presidents, and a five-person Advisory Board. The President cannot be Vice-President and cannot be on the Advisory Board. A Vice-President cannot be on the Advisory Board. Let $n$ be the number of students in Carleton's Computer Science program, where $n \geq 8$. In how many ways can a Board of Directors be chosen?

(a)

$(n-5){n \choose 2}{n-1 \choose 5}$

(b)

$(n-7){n \choose 2}{n-2 \choose 5}$

(c)

$(n-2){n\choose 2}{n-2 \choose 5}$

(d)

$n{n \choose 2}{n \choose 5}$

Solution

First, we choose 2 of the n students to be Vice-Presidents. There are $\binom{n}{2}$ ways to do this.

Then, we choose 5 of the remaining n-2 students to be on the Advisory Board. There are $\binom{n-2}{5}$ ways to do this.

Finally, we choose one of the remaining n-7 students to be the President. There are $\binom{n-7}{1}=(n-7)$ ways to do this.

Thus, the number of ways to choose the Board of Directors is $\binom{n}{2} \cdot \binom{n-2}{5} \cdot (n-7)$