AUSTRIAN-POLISH 1979 Problem 7

Let $n$ and $m$ be fixed positive integers. The hexagon $ABCDEF$ with vertices $A = (0,0)$ , $B = (n,0)$ , $C = (n,m)$ , $D = (n-1,m)$ , $E = (n-1,1)$ , $F = (0,1)$ has been partitioned into $n+m-1$ unit squares. Find the number of paths from $A$ to $C$ along grid lines, passing through every grid node at most once.