The suggested solution seems implying exactly 9 rolls but not 10...

Anyway, the bottom line is:

1. For the negative binomial, you are required that the last roll is success,

while the order of success in the previous rolls are not important, and hence

you get the binomial coefficient. So in your formula, you are getting the

7th success exactly on the 10th roll. And thus you miss some cases where

you get the 7th success before the 10th roll, which shall be included by the

question.

2. For the binomial distribution, the order of success in the 10 rolls are not

important. They can be appear in any order, but you just required that there

are 7 successes in total.