Solution: 2017 Winter Midterm - 1

Author: Michiel Smid

Question

Carleton's Computer Science program has $f$ female students and $m$ male students, where $f \geq 1$ and $f + m \geq 8$. The Carleton Computer Science Society has a Board of Directors consisting of a President, five Vice-Presidents, and two Cider-Presidents (whose task is to buy cider for the President). All members of the Board of Directors are Computer Science students. The President is female, cannot be Vice-President, and cannot be Cider-President. A Vice-President cannot be Cider-President. In how many ways can a Board of Directors be chosen?

(a)

$f \cdot {f+m-1 \choose 7}$

(b)

$f \cdot {f+m-1 \choose 5} \cdot {f+m-6 \choose 2}$

(c)

$f \cdot {f+m \choose 5} \cdot {f+m \choose 2}$

(d)

$f + {f+m-1 \choose 5} + {f+m-6 \choose 2}$

Solution

There are $ f $ amount of possibile peeps to choose to be president: $ f $

Excluding the person we picked to be president, we choose 5 people to be vice presidents from the remaining $ f+m-1 $ people: $ \binom{f+m-1}{5} $

Excluding the person we picked to be president and the 5 people we picked to be vice presidents, we choose 2 people to be cider presidents from the remaining $ f+m-6 $ people: $ \binom{f+m-6}{2} $

Thus, there are $ f \cdot \binom{f+m-1}{5} \cdot \binom{f+m-6}{2} $ ways to choose a Board of Directors.