TITLE: Approximating the logarithm of the binomial coefficient
QUESTION [24 upvotes]: We know that by using Stirling approximation:
$\log n! \approx n \log n$
So how to approximate $\log {m \choose n}$?

REPLY [4 votes]: If you expand stirling's approximation for $n\choose\alpha n$, you may also notice that it can be written as 
$$\log{n\choose\alpha n} = n H(a) - \tfrac12\log(2\pi n\,\alpha(1-\alpha)) + O\left(\tfrac1n\right)$$,
where
$$H(a) = \alpha \log\tfrac1\alpha + (1 - \alpha) \log\tfrac1{1-\alpha}$$
is the binary entropy function (in nats).
If you further notice that $n\alpha(1-\alpha)$ is the variance of a binomial distribution, it becomes pretty easy to remember. (And you start to see how you might derive the central limit theorem for binomial distributions).