TITLE: Prime $p$ does not divide $\dbinom{n}{p^e}$
QUESTION [5 upvotes]: I encountered this statement as a lemma in the course of proving the first Sylow theorem.
If $|G|=n=p^em$, $p$ is a prime and $m$ and $p$ are coprime, show that
$p$ does not divide $\dbinom{n}{p^e}$.
I tried expanding to factorials, but it didn’t work.
A solution or directive towards the proof is appreciated.

REPLY [2 votes]: It seems to me that nothing but the definition of the binomial coefficient is
needed for this.
Indeed,
by the definition of the binomial coefficient we have
$$
{n\choose p^e}=
\frac{n(n-1)(n-2)\ldots(n-p^e+1)}{1\cdot2\cdots(p^e-1)\cdot p^e}=
\frac{n}{p^e}\prod_{k=1}^{p^e-1}\frac{n-k}{k}.
$$
It remains to be seen that none of the factors of the last expression is divisible by $p$.