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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.

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2 Answers

up vote 3 down vote accepted

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.

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thank you! now i remember it, it's the Newton thing... :) –  athos Aug 30 '12 at 1:25
    
@athos: Yes, the discovery of the extended binomial was an important step in the development of the calculus. –  André Nicolas Aug 30 '12 at 1:30
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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.

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thank you! both answers are nice, unfortunately can pick only 1 answer.. –  athos Aug 30 '12 at 1:25
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Hey, no worries. The feedback is far more valuable than a little green tick! –  Fly by Night Aug 30 '12 at 1:29
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