Solution: 2017 Winter Midterm - 7

Author: Michiel Smid

Question

Let $m \geq 2$ and $n \geq 2$ be integers. What does $$ {m \choose 2} + {n \choose 2} + mn $$ count?

(a)

The number of ways to choose an ordered pair $(x,y)$ from a group consisting of $m$ men and $n$ women, where $x$ and $y$ cannot both be men.

(b)

The number of ways to choose an unordered pair of people from a group consisting of $m$ men and $n$ women.

(c)

The number of ways to choose an ordered pair $(x,y)$ from a group consisting of $m$ men and $n$ women, where $x$ must be a man and $y$ must be a woman.

(d)

The number of ways to choose an unordered pair of people from a group consisting of $m$ men and $n$ women, where at least one man must be chosen.

Solution

It’s a little confusing when you know why the correct answer is correct, but you don’t know why the incorrect answers are incorrect.

Let’s go through all of them

The number of ways to choose an ordered pair $(x,y)$ from a group consisting of $m$ men and $n$ women, where $x$ must be a man and $y$ must be a woman.
So this one is wrong because the phrase they use only takes into account when we have a man and a women: $m \cdot n$
But the equation given shows that we could have 2 men or 2 women as a unordered pair.
The number of ways to choose an ordered pair $(x,y)$ from a group consisting of $m$ men and $n$ women where $x$ and $y$ cannot both be men.
For this one, it says $x$ and $y$ cannot both be men but $ \binom{m}{2} $ gives us unordered pairs of men in the equation
Since the phrase doesn't match the equation, this one is wrong.
The number of ways to choose an unordered pair of people from a group consisting of $m$ men and $n$ women, where at least one man must be chosen.
For this one, it says \enquote{at least one man must be chosen} but the equation has $ \binom{m}{2} $ which gives us unordered of women as well
Since the phrase doesn't match the equation, this one is wrong.
The number of ways to choose an unordered pair of people from a group consisting of $m$
men and $n$ women.
For this one, the phrase matches the equation. We take into account 2 men, 2 women, and 1 man-1 woman