Inequality involving factorial $\binom nk<(en/k)^k$ 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 $$\binom{n}{k}\approx \frac{\sqrt{2\pi n}n^n}{2\pi \sqrt{k(n-k)}(n-k)^{n-k}k^k}$$

$$\binom{n}{k} \left( \frac{k}{en} \right)^k = \frac{n(n-1) \ldots (n-k+1)}{n^k} \frac{k^k}{k! e^k} \leq \frac{k^k}{k! e^k} \text{ and since } e^k = \sum_m \frac{k^m}{m!},\;\;\; \frac{k^k}{k! e^k} < 1.$$
