Sum $(1-x)^n$ $\sum_{r=1}^n$ $r$ $n\choose r$ $(\frac{x}{1-x})^r$ The question is to find the value of:
$n\choose 1$$x(1-x)^{n-1}$ +2.$n\choose2$$x^2(1-x)^{n-2}$ + 3$n\choose3$$x^3(1-x)^{n-3}$ .......n$n\choose n$$x^n$.
I wrote the general term and tried to sum it as:
S=$(1-x)^n$$\sum_{r=1}^n$$r$$n\choose r$$(\frac{x}{1-x})^r$.
I got stuck here.
What do I do after this?

HINT: $r\binom{n}r=n\binom{n-1}{r-1}$; this is easy to see if you expand into factorials, and it also has a straightforward combinatorial proof.
