How can I prove this binomial identity?

[Pietjuh]

Homework Statement

Prove that the following binomial identity holds:

$${n+k-1 \choose k} = \sum_{i=1}^k {k-1\choose i-1}{n\choose i}$$

The Attempt at a Solution

One of the methods I've tried is to use induction on the variable n, but while trying this I got stuk on rewriting the binomial coefficients... can someone give me a hint if I can use another simple binomial identity for this?

Another thing I have tried to do is to look at the generating function of the left hand side, and then try to rewrite this to a generating function for the right hand side, but that didn't succeed either...

Can someone point me a little bit in the right direction?

 

[morphism]

Try finding two ways to count the number of ways you can choose k balls from a box containing n red balls and k-1 red balls.

Also note that
$${k-1 \choose i-1} = {k-1 \choose k-i}$$