generalized identity explained

Last time I discussed the identity k*C(n,k) = n*C(n-1,k-1). This is a general case of the identity
C(n,k)*C(k,m) = C(n,m)*C(n-m,k-m). If you substitute m = 1 you get the identity above. The combinatorial interpretation of the second one is choose k people from n and then choose m executives from those m people. The right hand side is first choose m executives from n and then choose from the remaining people choose k-m so as to have total of k people.