Problem:
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 is 9 is?

Solution:
To find the probability that the largest number appearing on a selected coupon is 9, we need to consider the possible outcomes and the total number of outcomes.

Possible Outcomes:
Let's consider the possible outcomes for the largest number appearing on a selected coupon to be 9.

- The largest number appearing on a selected coupon can be 9, and the remaining 6 numbers can be any number from 1 to 9. So, the number of possible outcomes in this case is 9^6.
- The largest number appearing on a selected coupon can be any number from 10 to 15, and the remaining 6 numbers can be any number from 1 to 15. So, the number of possible outcomes in this case is 6 * 15^6.

Total Number of Outcomes:
The total number of outcomes is the total number of ways to select 7 coupons out of the given 15 coupons with replacement. This can be calculated using the formula for combinations with replacement:

Total number of outcomes = (n + r - 1) C r

where n is the number of items to choose from and r is the number of items to be chosen.

In this case, n = 15 (total number of coupons) and r = 7 (number of coupons to be selected). So, the total number of outcomes is (15 + 7 - 1) C 7.

Calculating the Probability:
To calculate the probability, we need to divide the number of possible outcomes by the total number of outcomes.

Probability = (Number of Possible Outcomes) / (Total Number of Outcomes)

Substituting the values, the probability that the largest number appearing on a selected coupon is 9 is:

Probability = (9^6 + 6 * 15^6) / ((15 + 7 - 1) C 7)

Simplifying the expression, we can calculate the probability.

Conclusion:
The probability that the largest number appearing on a selected coupon is 9 can be calculated using the formula for combinations with replacement.