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It's in the following form:

$ \int_0^1 r^k (1-r)^{n-k} dr $

I tried expanding the $(1-r)^{n-k}$ part, but it looks really complicated.

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    Read about the [Beta function](http://en.wikipedia.org/wiki/Beta_function) on wikipedia.2012-12-18

1 Answers 1

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This integral (with the constants slightly changed) is known as the beta function:

$\mathrm{B}(x,y) := \int_0^1t^{x-1}(1-t)^{y-1}\,dt = \dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}$

It has been very well studied and is important in integration and many other fields.

Your integral is equal to $\mathrm B(k+1,\, n-k+1)$.

http://mathworld.wolfram.com/BetaFunction.html

http://en.wikipedia.org/wiki/Beta_function