Let $h(p) = -p \log p-(1-p)\log (1-p)$ denote the binary entropy of a Bernoulli distribution when the probability of observing a zero is $p$, where $\log$ denotes the logarithm to base 2. Show, using Stirling's approximation:
$$ \ln(n!) = (n+\frac{1}{2})\ln(n) - n + O(1)$$
that the following holds:
$$\log\binom{n}{\gamma n} = nh(\gamma) -\frac{1}{2} \log n + O(1).$$
How should I do this? I'd like a hint to get me started.