Bipartite Graphs and Matchings · Challenge problem

$X=\{1,2,3,4\}$ and $Y=\{a,b,c,d\}$ are joined by every edge of $K_{4,4}$ except $1a$ and $2a$ (both $1$ and $2$ are forbidden from pairing with $a$; every other edge is present). How many perfect matchings does this graph have?