# How to prove elementary identities for binomial coefficients using combinatorial arguments?

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!

Sometimes a bit of lateral thinking can help to spot such an argument. The binomial $\binom{n}{k}$ is a way of counting the $k$-subsets of an $n$-set. Multiplying by $(n-k)$ is how many ways one could single out an element of the remaining $(n-k)$ items left after picking that $k$-subset.
So this is a bit like the "bonus ball" being chosen as the final draw in a lottery. The order of the first $k$ items doesn't matter; they are treated as a $k$-set. But the final ball drawn has a special significance, making it harder to earn the top prize by correctly identifying that number.
• This probably gives you an idea about proving some additional expressions are equal to these two. For example, suppose we start by picking $k+1$ balls, and then choose one of them to be the "bonus" ball? – hardmath Jan 31 '15 at 17:33