<p>In how many ways can 10 different prizes be given to 5 students if one particular boy must get 4 prizes and rest of the students can get any number of prizes?</p>
Step-by-Step Solution
Key Concept: One student is fixed to receive exactly 4 prizes from 10; the remaining 6 prizes can be distributed freely among all 5 students (including the fixed student). Use combinations for selection, then surjections or inclusion-exclusion for unrestricted distribution.
<p><strong>Step 1:</strong> Select 4 prizes for the particular boy from 10 prizes.</p><p>Number of ways = C(10,4) = 210</p><p><strong>Step 2:</strong> Distribute the remaining 6 prizes among 5 students with no restrictions (each prize can go to any of the 5 students independently).</p><p>Number of ways = 5^6 = 15,625</p><p><strong>Step 3:</strong> Apply the multiplication principle since these are independent events.</p><p>Total ways = C(10,4) × 5^6 = 210 × 15,625 = 3,281,250</p><p><strong>∴ Answer: 3,281,250</strong></p>
Correct Answer: 3