<p>Number of ways in which 25 identical things be distributed among five persons if each gets odd number of things is</p>
Step-by-Step Solution
Key Concept: If each person must get an odd number of items from a total of 25 (odd), represent each person's share as 2kᵢ + 1, then the constraint transforms into distributing identical items among persons with a simpler structure using substitution.
<p><strong>Step 1:</strong> Let person i receive aᵢ things where aᵢ must be odd for i = 1,2,3,4,5.</p><p><strong>Step 2:</strong> Since each aᵢ is odd, write aᵢ = 2bᵢ + 1, where bᵢ ≥ 0.</p><p><strong>Step 3:</strong> The constraint becomes: (2b₁ + 1) + (2b₂ + 1) + (2b₃ + 1) + (2b₄ + 1) + (2b₅ + 1) = 25</p><p><strong>Step 4:</strong> Simplifying: 2(b₁ + b₂ + b₃ + b₄ + b₅) + 5 = 25</p><p><strong>Step 5:</strong> Therefore: b₁ + b₂ + b₃ + b₄ + b₅ = 10, where each bᵢ ≥ 0</p><p><strong>Step 6:</strong> Using stars and bars, the number of non-negative integer solutions is C(10+5-1, 5-1) = C(14, 4)</p><p><strong>Step 7:</strong> C(14, 4) = (14 × 13 × 12 × 11)/(4 × 3 × 2 × 1) = 24024/24 = 1001</p><p>∴ Answer: C (1001)</p>
Correct Answer: C