Bipartite Graphs and Matchings · Superchallenge problem

$X=\{1,2,3,4,5\}$ and $Y=\{a,b,c,d,e\}$ are joined by every edge of $K_{5,5}$ except $1a,2b,3c,4d,5e$ AND $1b$ (six forbidden pairs total). How many perfect matchings does this graph have?