I have a set of questions like the following:

For the function $\displaystyle \left ( 1+x \right )\left ( 1-x \right )^{9}$, find the term containing $\displaystyle x^{r}$ and hence express the expansion in sigma notation.

The answer for this particular problem is given at the back of the book as

$\displaystyle \sum_{r=0}^{9}\left ( -1 \right )^{r}\frac{9!x^{r}\left ( 10-2r \right )}{r!\left ( 10-r \right )!}$

I get that there is some manipulation of the binomial coefficient, but I am wondering if there is a standard approach to these types of problems, and it confuses me that r still goes to 9 when there should now be 11 terms it seems.

Thoughts appreciated.

Andrew