Proving A Combination Statement By Forming committees

Question

$ \binom{n}{m}\binom{m}{m} + \binom{n}{m+1}\binom{m+1}{m} + \binom{n}{m+2}\binom{m+2}{m} + \cdots + \binom{n}{n}\binom{n}{m} = \binom{n}{m} 2^{n-m}$

I am completely lost on where to go. Any hints? Thank you very much!

Answer 1

There are $n$ people. From these, some will go to a party. From people that go to the party, $m$ will be singers.

Now you can either choose who go to party first (at least $m$), then from the chosen, you choose who will sing.

Or you can choose who will go to party to sing first, and then the remaining $n-m$ people may or may not go to party