<p>Fifteen coupons are numbered 1, 2, 3, . . . 15. In how many ways seven coupons are selected such that the largest number appearing on the selected coupon is 9?</p>
Step-by-Step Solution
Key Concept: If the largest selected coupon is 9, then coupon 9 must be selected, and the remaining 6 coupons must come from {1, 2, 3, ..., 8}. This reduces to selecting 6 coupons from 8 available coupons.
<p><strong>Step 1:</strong> Understand the constraint. We need the largest selected coupon to be exactly 9. This means:</p><ul><li>Coupon 9 MUST be selected</li><li>No coupon from {10, 11, ..., 15} can be selected</li><li>The remaining 6 coupons must come from {1, 2, 3, ..., 8}</li></ul><p><strong>Step 2:</strong> Since coupon 9 is mandatory, we only need to select 6 more coupons from the set {1, 2, 3, 4, 5, 6, 7, 8}.</p><p><strong>Step 3:</strong> Number of ways = C(8, 6) = C(8, 2) = (8 × 7)/(2 × 1) = 28</p><p>∴ Answer: D (28)</p>
Correct Answer: D