Move everything to one side and regroup:

This can be rewritten as

I'll let a = m-1, b = n-1, c = r-1 where a,b,c are non-negative integers, that is .

Here, either (a,b,c) = (1,1,1) or (0,1,3) (up to re-arranging). This leaves 1! + 3! = 7 solutions.