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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.


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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Perhaps you should have a look here:… – Martin Sleziak Sep 19 '11 at 15:43
up vote 2 down vote accepted

The easiest way is to use the recurrence identity from Pascal triangle

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

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