Binomial Coefficients – Approximating the Logarithm

Question

We know that by using Stirling approximation:
$\log n! \approx n \log n$

So how to approximate $\log {m \choose n}$?

Answer 1

A better approximation for the logarithm of a factorial can be found by using $\log n! \approx n \log n - n$. Interestingly, the additional terms in the approximation of the binomial coefficient cancel out, and the result is the same as if you used the simpler approximation $\log n! \approx n\log n$:

$$\begin{align} \log {n\choose m} & = \log \frac{n!}{m!(n-m)!} \\ & = \log n! - \log m! - \log (n-m)! \\ & \approx n \log n - n - m \log m + m - (n-m) \log (n-m) + n-m \\ & = n \log n - m \log m - (n - m) \log (n-m) \end{align}$$

An even better approximation uses more terms of Stirling's approximation, giving $\log n! \approx (n+\tfrac{1}{2})\log n - n + \tfrac{1}{2}\log 2\pi$ and hence

$$\begin{align} \log {n\choose m} &\approx (n+\tfrac{1}{2})\log n - (m+\tfrac{1}{2})\log m - (n-m+\tfrac{1}{2})\log (n-m) - \tfrac{1}{2}\log 2\pi \\ & = n\log n - m \log m - (n-m)\log(n-m) + \cdots \\ & \qquad +\tfrac{1}{2} (\log n - \log m - \log (n-m) - \log 2\pi) \end{align}$$

where the last term is the correction from using more terms of Stirling's approximation.