TITLE: Proving $e^{\binom{n}{2}}>n!$
QUESTION [6 upvotes]: Prove that $$e^{\binom{n}{2}}>n!$$
$n \in \mathbb{Z_+}$
Sorry, couldn't attempt it.

REPLY [2 votes]: I quite like Thomas Andrews' approach. Alternatively you can estimate
$$
\ln(n!)=\sum_{k=1}^n\ln k <\int_1^{n+1}\ln x\,dx.
$$
And calculating that integral gives you a good enough upper bound on the r.h.s. Admittedly this needs more machinery, but it also gives a better approximation to $n!$.