Consider a set of n boys and n girls. In each situation below, use inclusion-exclusion to derive formulas for the number of ways to partition the $2n$ people into pairs so that for each $i$ the $i$-th tallest boy is not paired with the $i$-th tallest girl. (keep in summation form)
(a) Same-sex pairs allowed.
(b) No same-sex pairs allowed.
I know that the Principle of Inclusion-Exclusion states that given sets $A_1,...,A_n$ we have that
$|\cup_{i=1}^n A_i| = \sum\limits_{S\subseteq[n]} (-1)^{|S|}|\cap_{i \in S} A_i|$
To group $2n$ people into distinct pairs, I would have to count $\frac{(2n)!}{2^nn!} = \prod\limits_{i=1}^n (2i-1)$ pairs (they are equivalent statements).
I don't know how to continue from here and use the summations exactly to do both the questions.