While I was reading Molecular Driving Forces second edition by Ken A. Dill, I was confused by how to find the derivative of the multiplicity function $$W(n,N) = \frac{N!}{n!(N-n)!}$$
in order to find n when W is at maximum I know it is a discrete function, so I tried to find answers here Gamma Function, but still couldn't figure it out. In the textbook, it only says by finding $\frac{dW}{dn}$, we can know the extremum is at $n=\frac{N}{2}$. Can anyone help me with this please?
You're right that to use calculus, one would need to interpolate these factorials by differentiable functions (perhaps using the Gamma function).
But it's much easier than that to determine the maximum value of $W(n,N)$ for a fixed $N$. Note that for any $0\le n\le N-1$, $$ \frac{W(n+1,N)}{W(n,N)} = \frac{N!/(n+1)!(N-n-1)!}{N!/n!(N-n)!} = \frac{N-n}{n+1}. $$ The right-hand side is greater than $1$ precisely when $N-n>n+1$, or equivalently when $n < (N-1)/2$. In other words, $W(n+1,N)$ exceeds $W(n,N)$ precisely when $n < (N-1)/2$. It follows immediately that the largest value of $W(n,N)$ is the first value of $n$ that is at least $(N-1)/2$, which is $\lceil(N-1)/2\rceil = \lfloor N/2\rfloor$. (When $N$ is odd, the two central values are tied for largest.)
