# Proving that $n \choose k$ is an integer [duplicate]

Possible Duplicate:
Proof that a Combination is an integer

I can't think how to prove that ${n\choose k} \in\mathbb{Z}$.

I've played with it for a while, using the factorial definition for ${n\choose k}$. Must be something to do with factors but I'm struggling to prove.

Thanks.

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## marked as duplicate by Mike Spivey, Ｊ. Ｍ., Chris Eagle, JavaMan, Byron SchmulandSep 19 '11 at 16:33

$$\binom{n+1}{k+1} = \binom{n}{k+1} + \binom{n}{k}$$ and notice that $\binom{0}{0} = 1$, $\binom{n}{0} = \binom{n}{n} = 1$ for $n \in \mathbb{N} \cup \{ 0 \}$.