I'm in a second year discrete mathematics course, and we have identities like this $$\binom{n}{k}(n-k) = \binom{n-1}{k}n$$ and Pascal's Triangle law.
Our professor said that algebraic proofs are fine (and I have them) but is encouraging us to learn combinatorial arguments. I have them for most the rest of the questions; however, this one is stumping me. I'm looking over my notes, and it looks most like $A_k^n$, but that hasn't really gotten my anywhere. I'm not sure how to parse the LHS and RHS into any meaningful counting argument. Any help would be much appreciated!