1. **Problem statement:** We have 15 coupons numbered 1 to 15. We select 7 coupons with replacement. We want the probability that the largest number appearing among the selected coupons is exactly 9. 2. **Understanding the problem:** The largest number is 9 means all selected coupons are from 1 to 9, and at least one coupon is 9. 3. **Total number of outcomes:** Since selection is with replacement, each draw has 15 possibilities, so total outcomes are $$15^7$$. 4. **Favorable outcomes:** - All selected numbers are from 1 to 9: $$9^7$$ outcomes. - All selected numbers are from 1 to 8 (largest less than 9): $$8^7$$ outcomes. 5. **Number of outcomes where largest is exactly 9:** $$9^7 - 8^7$$ (all from 1 to 9 minus all from 1 to 8). 6. **Probability:** $$\text{Probability} = \frac{9^7 - 8^7}{15^7}$$ 7. **Final answer:** $$\boxed{\frac{9^7 - 8^7}{15^7}}$$