Question: 2015 Winter Final - 6
Let $n \geq 7$ be an integer. Why does the following equality hold:
$$
{n \choose 5}{n - 5 \choose 2} = {n \choose 2}{n - 2 \choose 5}
$$
(a)
Because both sides count the number of pairs $(A,B)$ of subsets of the $n$ people, such that $|A| = 2$, $|B| = 5$, and $A \subseteq B$.
(b)
Because both sides count the number of ways to choose 2 committees in a group of $n$ people, one committee has 5 members, the other committee has 2 members, and a person can be on both committees.
(c)
Because both sides count the number of ways to choose 2 committees in a group of $n$ people, one committee has 5 members, the other committee has 2 members, and <strong>no</strong> person can be on both committees.
(d)
Because both sides count the number of pairs $(A,B)$ of subsets of the $n$ people, such that $|A| = 5$, $|B| = 2$, and $A \subseteq B$.