<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 3:</strong> If a person buys 8 such packets, then the probability that he gets two free packets is</p>
Step-by-Step Solution
Key Concept: To get exactly 2 free packets, the person must complete exactly 2 sets of {S,U,R,F} from 8 packets. This means organizing 8 letters into exactly 2 complete sets with optimal distribution.
<p><strong>Step 1:</strong> To get exactly 2 free packets means completing exactly 2 full sets of {S,U,R,F}. From 8 packets, this requires: at least 2 of each letter, with exactly 4 letters being 'extra' (beyond the minimum 2 of each).</p><p><strong>Step 2:</strong> Required distribution of 8 packets: Each of S, U, R, F appears at least 2 times. The remaining 8 - 8 = 0 packets... Actually, with 8 packets and 4 letters needing minimum 2 each = 8 packets exactly. So we need exactly 2 of each letter.</p><p><strong>Step 3:</strong> The probability of getting exactly 2 of each letter (S, U, R, F) in 8 packets is the multinomial coefficient divided by total outcomes:</p><p>P = (8!)/(2!·2!·2!·2!) ÷ 4^8 = (40320)/(2·2·2·2) ÷ 65536 = 2520/65536 = 315/8192</p><p><strong>Step 4:</strong> Simplifying: 315/8192 is in lowest terms.</p><p>∴ Answer: D (where D represents 315/8192 or equivalent simplified form)</p>
Correct Answer: D