Counting number of states for fermions

I have a system of $N$ fermions that can occupy $M$ single particle states, and $M$ is much larger than $N$, $M gg N$. Since only a single or no fermion can be in a particular state, the number of way of counting the number of states for indistinguishable fermions is going to simply be ${M choose N}$. This means that the number of states $n$, is
begin{align*}
n &= {Mchoose N}
&= frac{M!}{(M-N)!N!}
end{align*}
Using Stirling’s approximation,
begin{align*}
n &= frac{(M/e)^M}{left((M-N)/eright)^{M-N}(N/e)^N}
&= frac{M^M}{(M-N)^{M-N}N^N} = frac{M^M}{(1-N/M)^{M-N} cdot M^{M-N}N^N}
&= left(frac{M}{N} right) ^N cdot left( 1-frac{N}{M} right)^{N-M}
&implies boxed{n = left(frac{M}{N} right) ^N cdot left( 1-frac{N}{M} right)^{N-M} }
end{align*}

Does this process make sense? Am I using the approximation right?

Any advice would be appreciated.