Simplest proof that $\binom{n}{k} \leq \left(\frac{en}{k}\right)^k$

Question

The inequality $\binom{n}{k} \leq \left(\frac{en}{k}\right)^k$ is very useful in the analysis of algorithms. There are a number of proofs online but is there a particularly elegant and/or simple proof which can be taught to students? Ideally it would require the minimum of mathematical prerequisites.

Answer 1

$\binom{n}{k}\leq\frac{n^k}{k!}=\frac{n^k}{k^k}\frac{k^k}{k!}\leq\frac{n^k}{k^k}\sum_m\frac{k^m}{m!}=(en/k)^k$.

(I saw this trick in some answer on this site, but can't recall where.)