This is certainly a valid proof, but also is entirely useless. Even if you understand the proof perfectly, it does not tell you why the identity is true. A better approach would be to explain what \({n \choose k}\) means and then say why that is also what \({n-1 \choose k-1} + {n-1 \choose k}\) means. Let's see how this works for the four identities we observed above.
in-context