TITLE: Dividing factorials is always integer
QUESTION [5 upvotes]: Is there a simple way to show that $$n!\over r!(n-r)!$$ is always an integer?

REPLY [4 votes]: The symmetric group $S_n$ has $n!$ elements. Letting it act on $n$ points, of which $r$ are coloured red the the remaining  $n-r$ are coloured yellow, the subgroup of $S_n$ that of permutation that do no affect the colouring is isomorphism to $S_r\times S_{n-r}$ (permuting red and yellow points among each other independently). By Lagrange's theorem, the order $r!(n-r)!$ of the stabiliser subgroup divides the order $n!$ of the whole group.