Solution: 2019 Fall Midterm - 7
Author: Michiel SmidQuestion
What does
$$
{w \choose 3} \cdot {m \choose 2} + {w \choose 4} \cdot m + {w \choose 5}
$$
count?
(a)
The number of ways to choose 5 people out of a group consisting of $w$ women and $m$ men,
where at most 3 men can be chosen.
(b)
The number of ways to choose 5 people out of a group consisting of $w$ women and $m$ men,
where at least 3 women must be chosen.
(c)
The number of ways to choose 5 people out of a group consisting of $w$ women and $m$ men,
where at most 3 women can be chosen.
(d)
The number of ways to choose 5 people out of a group consisting of $w$ women and $m$ men,
where at least 3 men must be chosen.
Solution
We can break this into 3 cases to see what it counts.
Case 1: 3 women and 2 men
We choose 3 women from $ w $: $ \binom{w}{3} $
We choose 2 men from $ m $: $ \binom{m}{2} $
In total, there are $ \binom{w}{3} \cdot \binom{m}{2} $ possibilities.
Case 2: 4 women and 1 man
We choose 4 women from $ w $: $ \binom{w}{4} $
We choose 1 man from $ m $: $ \binom{m}{1} $
In total, there are $ \binom{w}{4} \cdot \binom{m}{1} $ possibilities.
Case 3: 5 women and 0 men
We choose 5 women from $ w $: $ \binom{w}{5} $
In total, there are $ \binom{w}{5} $ possibilities.
Adding it all up, we get $ \binom{w}{3} \cdot \binom{m}{2} + \binom{w}{4} \cdot \binom{m}{1} + \binom{w}{5} $