Wikipedia describes how to prove the original equality, which is formula (8), using Vandermonde's identity:

I am not sure if your line of thought can be successfully continued, but it often happens in proofs by induction that one has to generalize the induction hypothesis P(n), and Vandermonde's identity seems to be precisely this generalization.

Wikipedia gives an algebraic and a combinatorial proofs of Vandermonde's identity, but it can also be proved by induction. If we denote (*) above by , then we can to prove by induction on . Here, it is important to choose the induction hypothesis correctly. One option in proving is to fix arbitrary and and then use induction, i.e., to prove and . However, according to my calculations, proving requires the induction hypothesis for a different , namely, . Therefore, one should fix only and then prove and . This way one gets a stronger induction hypothesis that works for all .

It is also possible to not fix and use induction hypothesis . In general, it is safer to import the universal quantifiers into the IH rather than fixing them before the induction starts.