Fifteen coupons are numbered $1,2,\dots ,15$ respectively. Seven coupons are selected at random one at a time with replacement. The probability that the largest number on a selected coupon as $9$ is 

Let $p=$the probability that a selected coupon bears number $\le 9.$
$\Rightarrow p=\frac{9}{15}=\frac{3}{5}$

                   $n=$ number of coupons drawn with replacement 
and            $X=$the number of coupons bearing number $\le 9.$
Note that $X~B(n,p).$
Probability that the largest number on the selected coupons does not exceed $9$

            $=$  probability that all the coupons bear number $\le 9$
             $=P(X=7){=}^7{C}_7{p}^7={\left(\frac{3}{5}\right)}^7.$

Similarly, probability that largest number on the selected coupon is $\le 8$ is ${\left(\frac{8}{15}\right)}^7.$ Hence, probability of the required event $={\left(\frac{3}{5}\right)}^7-{\left(\frac{8}{15}\right)}^7.$