I'm having some trouble proving the relation $$\frac{n!}{(n-k)!} = n^{\underline{k}}$$
Do you have to get into using gamma functions in order to prove this rigorously? Also, wikipedia seems to indicate this is an exact relation, while the thermodynamics books I'm learning out of has a question where it says the two are equal in the limit that $k \ll n$. And finally, why is the $k$ underlined? Thanks for the help!
The notation you are referring to is called the falling factorial. For integer $k$ it is defined as $n^{\underline{k}} = n(n-1)\ldots(n-k+1)$. It follows immediately that this equals $\frac{n!}{(n-k)!}$ (both are the product of integers from $n-k+1$ to $n$.