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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.

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up vote 8 down vote accepted

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

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

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Maybe it was here =]. –  Ragib Zaman Sep 25 '13 at 11:53
    
@RagibZaman So I actually saw it in several places :) –  user8268 Sep 25 '13 at 12:10

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