Proving $\sum\limits_{k=1}^n \binom{n}{k}\binom{n}{k-1} = \binom{2n}{n+1}$ I'm thinking about using a group of $2n$ balls and pick $n+1$ of them for the right hand side.
However, I have no idea how to do the left hand side. Or do I need to use induction?

In how many ways can we choose $n+1$ people from $n$ men and $n$ women? The RHS is obviously right, or we could pick which $k$ men qualify and which $k-1$ don't, giving the desired LHS.
