(a choose c)(b choose c) = sum_n (n choose n-a,n-b,c,a+b-n-c) (-1)^(a+b-n-c)
(where (x choose y, z, ... w) is the multinomial coefficient x!/(y!z!...w!) where it must be that y+z+...+w = x for the notation to make sense)
It sure seems to be true for a half-dozen small expamples I tried, but I can't get any traction trying to prove it in any intuitively sensible way.
