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Prove that: $p \,\,\left|\, {p \choose k} \right., \quad 0< k \lt p$ if $p$ is prime.

how to prove that with direct proof?

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    Combinatorially, the set ${X\choose k}$ of $k$-subsets of an $n$-element set $X$ (1) can be partitioned into $n$ disjoint equal-sized cells. The cells are $\Gamma_x=\left\{V:x\in V\in{X\choose k}\right\}$ for $x\in X$, which we may intuitively expect to be a partition by considering symmetry.2012-11-28

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Write out what the binomial coefficient is: $ {p\choose k}=\frac{p!}{k!(p-k)!}. $ $p$ divides the numerator since it has a factor of $p$, but $p$ can't divide the denominator because it is the product of integers smaller than $p$ and $p$ is prime.

This means that $p$ does not appear in the prime factorisation of the denominator, thus you can't simplify the $p$ factor that is on the numerator.

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    Glad to be of use2012-11-28