Have you ever seen Pascal's identity? It's a corollary to the Binomial Theorem. It says
This is restating in terms of combinations that the elements of Pascal's triangle are determined by the two elements directly above it. Anyways, if we use this identity on C(n,k), then on C(n-1,k-1) and C(n-1,k) we should get what we want.
Does this get things going in the right direction? Let me know if anything is confusing. Good luck!