<p>A committee of 10 persons is to be selected from 9 women and 8 men, consisting of at least 4 women and at least 4 men. Find the number of ways the committee can be formed if Miss X refuses to work with Mr. Y.</p>
Step-by-Step Solution
Key Concept: Use the principle of exclusion: calculate total valid committees and subtract those containing both Miss X and Mr. Y.
<p><strong>Step 1:</strong> Total valid committees (without restriction) with at least 4 women and 4 men from 10-person committee:</p><p>Possible distributions: (4W,6M), (5W,5M), (6W,4M)</p><p>\[\text{Total} = \binom{9}{4}\binom{8}{6} + \binom{9}{5}\binom{8}{5} + \binom{9}{6}\binom{8}{4}\]</p><p>\[= 126 \times 28 + 126 \times 56 + 84 \times 70 = 3528 + 7056 + 5880 = 16464\]</p><p><strong>Step 2:</strong> Subtract committees with both Miss X and Mr. Y:</p><p>If both X and Y are selected, we need 8 more from 8 women and 7 men, with constraint of at least 3 more women and 3 more men.</p><p>Distributions: (3W,5M), (4W,4M), (5W,3M)</p><p>\[\text{Bad} = \binom{8}{3}\binom{7}{5} + \binom{8}{4}\binom{7}{4} + \binom{8}{5}\binom{7}{3}\]</p><p>\[= 56 \times 21 + 70 \times 35 + 56 \times 35 = 1176 + 2450 + 1960 = 5586\]</p><p><strong>Step 3:</strong> Answer = 16464 - 5586 = 10878</p><p>∴ Answer is <strong>(D) 10878</strong></p>
Correct Answer: C