I am trying to solve a statistics problem and arrive at the following finite summation which stumps me but which Mathematica says evaluates to a simple closed form:
$$\sum_{k=0}^n(1-x)^{k+1}\frac{n!}{k!(n-k)!} = (1-x)(2-x)^n$$
I can't figure out how to prove this, I tried induction but was unsuccessful. Could somebody help me understand how to prove this?
By the binomial theorem $$(2-x)^n=(1+(1-x))^n=\sum_{k=0}^n(1-x)^k\binom{n}{k}.$$