Combinatorial Meaning of the Binomial Coefficient $s\choose k$?

Question

I am wondering if there is a combinatorial meaning of the binomial coefficient $s\choose k$, which is defined as $${s\choose k}=\frac{s(s-1)(s-2)\cdots(s-k+1)}{k!},$$ where $s$ is a real number and $k$ is a natural number. If $s$ is a natural number which is at least $k$, there is an obvious combinatorial meaning of $s\choose k$. Am wondering the general case where $s$ is a real number. Does $s\choose k$ still have any combinatorial meaning at all?