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I'm trying to prove the following identity, but I'm not very lucky in finding the proof:

[tex]

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

[/tex]

I've tried to interpret this as a combinatorial problem, and I know the left hand side is the number of k-repeatingcombinations (don't really know the english word for it. The dutch word is k-herhalingscombinaties). But how i can transfer this to the sum on the right hand side...

Can someone help me a bit with this?