Bounds for $\binom{n}{cn}$ with $0 < c < 1$. Are there really good upper and lower bounds for $\binom{n}{cn}$ when $c$ is a constant $0 < c < 1$?  I know that $\left(\frac{1}{c^{cn}}\right) \leq \binom{n}{cn}  \leq \left(\frac{e}{c}\right)^{cn}$.

Hint: as $n$ goes to infinity, you can approximate $\binom{n}{cn}$ by Entropy function as follows:
$$
\binom{n}{cn}\approx e^{nH(c)}.
$$
where $H(c)=-c\log(c)-(1-c)\log(1-c)$
