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.
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.
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 $\mathrm B(k+1,\, n-k+1)$.