TITLE: $m!n! < (m+n)!$ Proof?
QUESTION [29 upvotes]: Prove that if $m$ and $n$ are positive integers then $m!n! < (m+n)!$
Given hint:

$m!= 1\times 2\times 3\times\cdots\times m$ and $1<m+1, 2<m+2, \ldots , n<m+n$

It looks simple but I'm baffled and can't reach the correct proof. 
Thank you.

REPLY [16 votes]: One-line proof (some details omitted): ${m+n \choose m} > 1$ if $0 < m < n$.