<p><strong>For Problems 1–3</strong><br>A shopping mall is running a scheme: Each packet of detergent SURF contains a coupon which bears letter of the word SURF, if a person buys at least four packets of detergent SURF, and produce all the letters of the word SURF, then he gets one free packet of detergent.</p><p><strong>Problem 1:</strong> If a person buys 8 such packets at a time, then the number of different combinations of coupon he has is</p>
Step-by-Step Solution
Key Concept: We need to find the number of ways to distribute 8 identical packets among 4 distinct letters (S, U, R, F) where each letter must appear at least once. This is a stars and bars problem with the constraint that each letter gets minimum 1 coupon.
<p><strong>Step 1:</strong> Identify the constraint. A person buys 8 packets and must get all 4 letters (S, U, R, F). We need combinations of coupons, not their arrangements.</p><p><strong>Step 2:</strong> Let x₁, x₂, x₃, x₄ represent the number of times letters S, U, R, F appear respectively. We need: x₁ + x₂ + x₃ + x₄ = 8, where each xᵢ ≥ 1 (each letter appears at least once).</p><p><strong>Step 3:</strong> Apply change of variables: Let yᵢ = xᵢ - 1, so yᵢ ≥ 0. Then: (y₁ + 1) + (y₂ + 1) + (y₃ + 1) + (y₄ + 1) = 8, which gives y₁ + y₂ + y₃ + y₄ = 4.</p><p><strong>Step 4:</strong> Use stars and bars formula: The number of non-negative integer solutions is C(4 + 4 - 1, 4 - 1) = C(7, 3) = 35.</p><p>∴ Answer: D (35)</p>
Correct Answer: D