I found this very interesting upper and lower bound for the binomial coefficient that is taken from a book that I have not heard of.
Could someone help me understand how it is derived?
Here is the inequality:
Let:
- $k>1,n>k$ be integers
- $h(x) = -x\ln(x) - (1-x)\ln(1-x)$
Then:
$$\left(\sqrt{\frac{n}{8k(n-k)}}\right)e^{nh(k/n)} \le {n \choose k} \le \left(\sqrt{\frac{n}{2\pi{k}(n-k)}}\right)e^{nh(k/n)}$$
I understand that it comes from Stirling's Approximation and the binary entropy function with the details found in this book.
If anyone can offer tips on why this inequality holds, that will be very helpful.