# Product of consecutive integers as alternating sum

In the derivation of the Indian Buffet Process (Indian Buffet Process paper -- see page 1218), we have the following step: $\prod_{k=1}^{K_+}{(K-k+1)} = K^{K_+}-\frac{(K_{+}-1)K_{+}}{2}K^{K_{+}-1}+...+(-1)^{K_{+}-1}(K_{+}-1)!K$.

I can see why this would be true, but I don't know if this theorem has a name, and I also don't know how to prove it. Either would be appreciated!

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It seems like you should be able to show this by induction on $K_+$... –  Rafe Kettler Nov 5 '13 at 0:03