Proving this binomial identity

[chaotixmonjuish]

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


I'm not sure how to prove it, I understand the combinatorial proof..i.e. putting it to an example...but i can't derive one side and get the other.

 

[qntty]

How do you understand the combinatorial proof but not know how to prove it?

 

[chaotixmonjuish]

It was an example given in the book. I just don't see how they manipulated the binomial to move the stuff around.

 

[chaotixmonjuish]

Suppose you had a bag full of n balls. Suppose out of the n balls you had m green ones. Would the right hand side be adding up the ways to count all the gree balls and non-green balls. I didn't even really understand this reasoning. This identity is really frustrating me.