I found a question while I was trying to practice Combinatorics and Probabilistic methods.I tried to solve it with no success.. this is the question:
Use the Stirling approximation of the factorial to show that for every $0\leq p \leq 1$ there holds
$$\lim _{n\to \infty}\frac{1}{n}\log\binom{n}{pn}=H(p)$$ where $H(p)=-p\log(p) -(1-p)\log(1-p)$ is the binary entropy function.
Any help?