# Applying multiplication principle to counting subsets

In my textbook MP is strictly reserved to counting lists. Does what I do below to count subsets work?

Consider $3$ men(Ace, Bob, Corry) and $3$ women(Ann, Beth, Candace). Suppose we need to choose a team with $2$ men and $2$ women in it. An example team is $\{\text {Ace, Ann, Beth, Bob}\}$ which is the union of $\{\text {Ace, Bob}\}\{\text {Beth, Ann}\}.$ So we simply count $2-$lists whose first element is a set of two men and whose second element is a set of two women. We get the same number of $2-$lists if their first element is a set of two women. There are ${3 \choose 2} = 8$ two-men subsets and that many two-women subsets. So there are $8$ choices for the first element and $8$ choices for the second one. In all there are $8^2 = {3 \choose 2}^2 = 64$ two element lists = teams with two men and two women in each.

it's all good except $\binom 32=3$ (not 8) so the true answer should be 9.
• How many $5$-letter words are there with exactly two vowels? We could count $2-$lists of the form $(\{b, c, d\}\{a, e\})$. Then we get $\binom {21}{3} \binom{5}{2}$ which is wrong. Why does the approach above not work with this problem? – keys Apr 26 '15 at 21:42