Prove this using a combinatorial arguement.
First off, the right hand side can be argued like so. Suppose you have n republicans and n democrats. If you pick one party, you have n ways to select a leader for a committee. Then you have 2n-1 people left to form the committee from of a size n-1? Is this the right reasoning.
As for the left side, suppose you picked the size of the committee k, you have ways to choose the democrats and again with the republicans.