<p>To form a committee of 11 persons from 8 males and 5 females, find the number of ways \(m\) to form a committee with at least 6 males. Also find the number of ways \(n\) to form a committee with at least 3 females. What is the value of \(m\) and \(n\)?</p>
<p>\(m = n = 78\)</p>
<p>\(m = n = 68\)</p>
<p>\(m = 78, n = 68\)</p>
<p>\(m = 68, n = 78\)</p>
Step-by-Step Solution
Key Concept: Break the constraint into mutually exclusive cases based on the gender composition. For 'at least 6 males' in an 11-person committee from 8 males and 5 females, enumerate: (6M,5F), (7M,4F), (8M,3F). Similarly for 'at least 3 females': (3F,8M), (4F,7M), (5F,6M). Calculate each case separately using combinations and sum them.
<p><strong>Step 1: Find m (at least 6 males from 8M, 5F, total 11 persons)</strong></p><p>If we have at least 6 males, we have at most 5 females. Valid cases:</p><p>• <strong>Case 1:</strong> 6 males, 5 females: C(8,6) × C(5,5) = 28 × 1 = 28</p><p>• <strong>Case 2:</strong> 7 males, 4 females: C(8,7) × C(5,4) = 8 × 5 = 40</p><p>• <strong>Case 3:</strong> 8 males, 3 females: C(8,8) × C(5,3) = 1 × 10 = 10</p><p><strong>m = 28 + 40 + 10 = 78</strong></p><p><strong>Step 2: Find n (at least 3 females from 8M, 5F, total 11 persons)</strong></p><p>If we have at least 3 females, we have at most 8 males. Valid cases:</p><p>• <strong>Case 1:</strong> 8 males, 3 females: C(8,8) × C(5,3) = 1 × 10 = 10</p><p>• <strong>Case 2:</strong> 7 males, 4 females: C(8,7) × C(5,4) = 8 × 5 = 40</p><p>• <strong>Case 3:</strong> 6 males, 5 females: C(8,6) × C(5,5) = 28 × 1 = 28</p><p><strong>n = 10 + 40 + 28 = 78</strong></p><p><strong>∴ m = 78 and n = 78</strong></p>
Correct Answer: A