  In this paper we prove several inequalities for binomial coefficients. For
instance, if $ k$ and $n$ are positive integers such that $n\ge 400$ and
$[\frac n5]\le k\le [\frac n2]$, where $[x]$ is the greatest integer not
exceeding $x$, then $$\binom nk<\Big(1-\frac{5(k-[\f n5])}{6n^2}\Big)
\frac{n^{n-\f 12}}{k^k(n-k)^{n-k}}.$$
