Combinatorial Proof I have trouble coming up with combinatorial proofs. How would you justify this equality?
$$
n\binom {n-1}{k-1} = k \binom nk
$$
where $n$ is a positive integer and $k$ is an integer.

We have a group of $n$ people, and want to count the number of ways to choose a committee of $k$ people with Chair. 
For the left-hand side, we select the Chair first, and then $k-1$ from the remaining $n-1$ to join her.
For the right-hand side, we choose $k$ people, and select one of them to be Chair. 
