what is the usage of combination $C(r,k)$ where $r$ extends to real number? Combination is defined as $C(n,k) = \dfrac{n!}{k!(n-k)!}$, where $n$ and $k$ are non-negative integers.
Now, the definition can be extended to $C(r,k)$, where $r$ is real number and $k$ is an integer:
$$
C(r, k) = \cases{
\frac{r(r-1)\cdots(r-k+1)}{k!} & $k \ge 0$ \\
1 & $k = 0$
\\ 0 & $k < 0 $}
$$
Question: what is the usage, or application cases, that such definition extension could help? I just don't see the real number $r$'s usage.
 A: One important use for $C(r,k)$, where $r$ is not a non-negative integer, is the  Generalized Binomial Theorem that, for $|x|\lt 1$, gives us the power series expansion for $(1+x)^r$.
The cases $r$ a negative integer and $r=1/2$ or $r=-1/2$ are particularly useful in combinatorics and elsewhere. The expansion of $(1+x)^{-1/2}$ is of special historical significance, since it leads immediately to a series expression for $(1-t^2)^{-1/2}$, and therefore (by term by term integration) to the $\arcsin$ function. 
A: You could, if you so wished, extend it over the complex numbers too. The $\Gamma$ function:
$$\Gamma(z) := \int_0^{\infty} e^{-t} \, t^{z-1} \, dt \, , $$
is a generalisation of the factorial function. It was defined by Euler and it has the property that $\Gamma(n) = (n-1)!$ for all integers $n \ge 1.$ The integral definition does not make sense for all $z \in \mathbb{C}$, but you can take the analytic continuation to give a function defined on all of $\mathbb{C}$ with only isolated poles. Although, to be honest, you'd probably need a computer algebra package to use it.
