# How do I evaluate the sum $\sum_{j=0}^d (-1)^j n^j \binom{d}{j}$

Let $n>0$ be an integer. Let $d$ be a positive integer.

How do I show that $$\sum_{j=0}^{2d} (-1)^j n^j \binom{2d}{j} = (n-1)^{2d}?$$

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This is simply $(1-n)^{2d}$ by the Binomial theorem. –  Ragib Zaman Feb 4 '12 at 11:41
You're completely right. –  Ali Feb 4 '12 at 11:54
Tag algebraic geometry? abelian varieties? –  GEdgar Feb 4 '12 at 14:36
Isn't this just the binomial expansion of $(1-n)^{2d}$?