Physical meaning of n!/k!

Question

Assume $n,k \in \mathbb{N}$ and $k < n$. Does $\frac{n!}{k!}$ have any physical meaning?

What I mean is this, $\frac{n!}{k!(n-k)!}$, can be interpreted as $n$ choose $k$ is there any physical interpretation of $\frac{n!}{k!}$.

Answer 1

The difference between $\dfrac{n!}{k!}$ and $\dfrac{n!}{k!(n-k)!}$ is whether we mind order of selection or not (the former one is often referred to as permutation). For example let we have a deck of cards. The number of cases to draw out four spades from spades and hearts without considering the order of selection is $\dfrac{26!}{4!22!}$ e.g. the cases $A\spadesuit,Q\spadesuit,7\spadesuit,2\spadesuit$ and $Q\spadesuit,2\spadesuit,A\spadesuit,7\spadesuit$ are the same but if considering these two latter cases as different the number of cases would be $\dfrac{26!}{22!}$