Find the probability the largest number appearing on the coupon is 9.

$9^7$ are the possibles 7-uples, but you want that at least one is 9. The competent of this is the 7-uples such that all elements aren't 9, there are $8^7$

There are $8^7$ combinations in which largest number appearing on a coupon is at most $8$.

There are $9^7$ combinations in which largest number appearing on a coupon is at most $9$.

Hence there are $9^7-8^7$ combinations in which largest number appearing on a coupon is $9$.

The point is that $9^7$ is the ways in which you can obtain $7$ numbers between $1$ and $9$, but since $9$ must be the max, you should also avoid all the cases in which all the numbers are from $1$ to $8$, making the max smaller than $9$, hence $9^7-8^7$.