Solution: 2016 Fall Midterm - 1

Author: Michiel Smid

Question

Carleton's Computer Science program has $f$ female students and $m$ male students, where $f \geq 1$ and $f + m \geq 4$. The Carleton Computer Science Society has a Board of Directors consisting of a President and three Vice-Presidents, all of whom are Computer Science students. The President is female and cannot be a Vice-President. In how many ways can a Board of Directors be chosen?

(a)

$f \cdot {f + m - 1 \choose 3}$

(b)

$(f - 1) \cdot {f + m \choose 3}$

(c)

$f \cdot {f + m \choose 3}$

(d)

${f + m \choose 4}$

Solution

First, we pick a female to be the President. There are $f$ ways to do this.

Then, we pick 3 of the remaining females-1(subtracted 1 because 1 of them is already president) and ales to be vice presidents.

There are $\binom{m-1+f}{3} \cdot m$ ways to do this.