Determine the number of perfect pairings in a complete bipartite graph Kn,n, in which n disjoint edges are removed.

Let's say our vertices are $a_1, \dots, a_n, b_1, \dots, b_n$. Since we are removing $n$ disjoint edges, up to relabeling we can remove the edges $(a_i, b_i)$ for $1 \leq i \leq n$.

We claim that the number of perfect matchings is the number of permutations of $\{1, \dots, n\}$ with no fixed points. Indeed, if we define $\sigma \in S_n$ as $\sigma(i) = j$ if and only if $(a_i, b_j)$ is an edge in the pairing, the fact that it is a perfect matching tells us that $\sigma$ is well defined (there is a unique edge containing each of the $a_i$ and each of the $b_j$), and the fact that we removed the pairs $(a_i, b_i)$ tells us that $\sigma(i) \neq i$ for all $i$. Vice versa, given any such permutation, we can define a perfect matching by taking the edges $(a_i, b_{\sigma(i)})$ for all $i$.

The number of permutations with no fixed points is well known and can be expressed in several ways, see https://en.wikipedia.org/wiki/Derangement for details, and the section https://en.wikipedia.org/wiki/Derangement#Derivation_by_inclusion%E2%80%93exclusion_principle for an explicit computation.