7
$\begingroup$

I am trying to prove following inequality:

$$\binom{n}{k}<(en/k)^k$$

I tried Stirling approximation but I could not get anything further. Then I get $$\binom{n}{k}\approx \frac{\sqrt{2\pi n}n^n}{2\pi \sqrt{k(n-k)}(n-k)^{n-k}k^k}$$

  • 0
    You may even be able to get the much sharper $\tbinom{n}{k}2011-06-02
  • 2
    @Ross: For $n=10$ and $k=3$, ${n \choose k}=120$ but $e(n/k)^k < 101$.2011-06-02

1 Answers 1