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Let $a_1, a_2, a_3,...,a_n$ be $n$ positive integers and if $a=a_1+a_2+a_3+...+a_n$. Prove that $a_1!a_2!a_3!...a_n! | a!$

Factorial of a sum of $n$ numbers is divisible by product of factorial of these numbers.


marked as duplicate by Community Oct 3 '17 at 16:16

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    $\begingroup$ Hint: how many ways can we permute a collection of $a_1$ apples, $a_2$ oranges, and so on. $\endgroup$ – lulu Oct 3 '17 at 15:57

This is the definition of Multinomial Theorem,

$(x_1 +x_2+\cdots+x_m)^n = \sum \limits_{k_1 +k_2 +\cdots +k_m =n} \binom{n}{k_1,k_2,\cdots,k_m} \prod \limits_{t=1}^{m} x_t^{k_t}$,

Its easy to see that $\binom{n}{k_1,k_2,\cdots,k_m}$ is Integer.


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