Solution: 2019 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)

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

(b)

${f + m \choose 3}$

(c)

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

(d)

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

Solution

We choose 1 of the $f$ girls to be the president: $\binom{f}{1}=f$

We choose 3 of the remaining $f-1$ girls and m guys to be vice presidents: $\binom{f-1+m}{3}$

Thus, the total number of ways to choose the president and vice presidents is $f \cdot \binom{f-1+m}{3}$