Question: 2022 Winter Final - 5
Let $S$ be a set of $m+w$ people, $m\ge 10$ of which are men and $w\ge 10$ of which are women. What does
\[
\sum_{k=2}^{10} \binom{m}{m-k}\cdot\binom{w}{10-k}
\]
count?
(a)
The number $10$-element subsets of $S$ that include at least $2$ women?
(b)
The number subsets of $S$ of size at most $10$ that include at least $2$ men?
(c)
The number subsets of $S$ of size at most $10$ that include at least $2$ women?
(d)
The number $10$-element subsets of $S$ that include at least $2$ men?
(e)
None of the Above