<p>The number of ways of choosing a committee of two women and three men from five women and six men, if Mr. A refuses to serve on the committee if Mr. B is a member and Mr. B can only serve, if Ms. C is the member of the committee is</p>
Step-by-Step Solution
Key Concept: Break the problem into mutually exclusive cases based on the conditional constraints: Case 1 (neither A nor B), Case 2 (A but not B), Case 3 (B with C, but not A). The key is recognizing that B's service depends on C, creating a dependency chain.
<p><strong>Step 1:</strong> Identify the constraints:</p><ul><li>Choose 2 women from 5 and 3 men from 6</li><li>Mr. A refuses if Mr. B is on committee (A and B mutually exclusive)</li><li>Mr. B can serve only if Ms. C is on committee (B requires C)</li></ul><p><strong>Step 2:</strong> <strong>Case 1 — Neither A nor B on committee:</strong></p><p>Women: Choose 2 from {5 women} = C(5,2) = 10</p><p>Men: Choose 3 from {6 − 2 (A, B)} = C(4,3) = 4</p><p>Case 1 total = 10 × 4 = 40</p><p><strong>Step 3:</strong> <strong>Case 2 — A is on committee, B is not:</strong></p><p>Women: Choose 2 from {5 women, Ms. C must be included} = C(4,1) = 4</p><p>Men: A is selected, need 2 more from {6 − 2 (A, B)} = C(4,2) = 6</p><p>Case 2 total = 4 × 6 = 24</p><p><strong>Step 4:</strong> <strong>Case 3 — B is on committee (C must be included), A is not:</strong></p><p>Women: Ms. C must be selected, choose 1 more from {4 remaining} = C(4,1) = 4</p><p>Men: B is selected, need 2 more from {6 − 2 (A, B)} = C(4,2) = 6</p><p>Case 3 total = 4 × 6 = 24</p><p><strong>Step 5:</strong> Total = 40 + 24 + 24 = <strong>88</strong></p><p>∴ Answer: C</p>
Correct Answer: C