How can I give a combinatorial proof for the identity C(n, r)C(r, k) = C(n, k)C(n - k, r - k)?
I have tried as follows:
Let S be a set with n elements, we want to choose a subset sub_1 of k elements and another disjoint subset of r - k elements, sub_2.
For the RHS we first choose sub_1 with C(n, k) and then sub_2 with C(n - k, r - k).
For the LHS we first choose sub_2 with C(n, r) and then sub_1 with C(r, k).
Is this correct at all? What am I doing wrong? It is really trivial to see algebraic but I find the combinatorial proofs hard.