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Does $p^k$ divide ${}_n\!C_r$ for all integer r if $p^k|n$ where $0\leq r \leq n$ and $p$ is prime?

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    I think it's $\binom{n}{r}$ (the number of $k$-combinations of $n$/the binomial coefficient).2012-07-05

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The answer is no. Take $p=3$, $n=6$, $k=1$, $r=3$. Then $p^k\mid n$ , but $p^k\nmid{}_n C_r$

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    Or $2^2\nmid{4\choose2}$. Or pretty much anything with $r=0$ or with $r=n$.2012-07-06