H(n-1, r)-H(n, r) = (r-1)!(n-r-1)! (n-1)! - (r-1)!(n-r)! n! = (r-1)!(n-r-1)! (n-1)! (1 - n-r n ) = (r-1)!(n-r-1)! (n-1)! ( r n ) = r! (n-r-1)! n! = H(n,r+1)