Solution: 2015 Winter Midterm - 1

Author: Michiel Smid

Question

The Carleton Computer Science Society has a Board of Directors consisting of a President and a seven-person Advisory Board. The 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)

$n \cdot {n \choose 7}$

(b)

$n + {n \choose 7}$

(c)

$(n - 7) + {n \choose 7}$

(d)

$(n - 7) \cdot {n \choose 7}$

Solution

First, we choose 7 of the n students.

Then, we choose one of the remaining n-7 students to be the President.

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