A store has n different products for sale. Each of them has a different price that is at least one dollar, at most n dollars, and is a whole dollar. A customer only has the time to inspect k different products. After doing so, she buys the product that has the lowest price among the k products she inspected. Prove that on average, she will pay $\displaystyle \frac{n+1}{k+1}$ dollars.

My attempt:

There are $\displaystyle ^nC_k$ ways to select k products out of n. The maximum amount she will ever pay is n-k+1.

There are $\displaystyle ^{n-1}C_{k-1}$ ways in which she ends up paying 1 dollar.

There are $\displaystyle ^{n-2}C_{k-1}$ ways in which she ends up paying 2 dollars.

... and so on.

Average = $\displaystyle \frac{1*^{n-1}C_{k-1}+2*^{n-2}C_{k-1}+...+(n-k+1)*^{k-1}C_{k-1}}{^nC_k}$

How do I proceed?