组合数求余

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Allen Hatcher, "Algebraic Topology", Lemma 3C.6.

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Lucas定理$\displaystyle \binom{n}{k} \equiv \prod_i\binom{n_i}{k_i} \bmod p$可以推广到multinomial${\displaystyle {n \choose k_{1},k_{2},\ldots ,k_{m}}={\frac {n!}{k_{1}!\,k_{2}!\cdots k_{m}!}}}$吗？

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