HELP PLZ

Neat problem.

First off note that $\dbinom{n}{k}=\dbinom{n}{n-k}$

We are choosing (n-1) items from 2n possible items.  One way to do this is to split the 2n items into 2 piles of n items each, choose k items from the first pile, and (n-1-k) items from the second pile.  We do this for $0 \leq k \leq n-1$

This gets us

$\dbinom{2n}{n-1} = \sum \limits_{k=0}^{n-1}~\dbinom{n}{k}\dbinom{n}{n-k-1} = \sum \limits_{k=0}^{n-1}~\dbinom{n}{k}\dbinom{n}{k+1}$

you can recognize the right hand term as the summation shown