Question

Fifteen coupons are numbered $1$ to $15$. Seven coupons are selected at random, one at a time with replacement. The probability that the largest number appearing on a selected coupon be $9$ is

• A. ${\left(\frac{1}{15}\right)}^7$
• B. ${\left(\frac{8}{15}\right)}^7$
• C. ${\left(\frac{3}{5}\right)}^7$
• D. none of these

Answer 1

The correct option is D none of these

Number of coupons =15
7 coupons are selected with replacement . So, total ways $={15}^7$
Now, for 9 to be the largest in selected coupons, so coupons should be selected from $1,2,...9$
Out of these 9 coupons, in 7 chances, we can select coupons in $9^7$ ways.
But this will also include cases when 9 is not selected at all, and thus, the highest number that is selected is 8. This can be done in $8^7$ ways (i.e. 8 coupons in 7 chances).
Thus, favourable cases for 9 to be largest in the selected coupon, number of cases $=9^7-8^7$
Required probability $=\frac{9^7-8^7}{{15}^7}$