<p>To form a committee of 11 persons from 8 male candidates and 5 female candidates, the number of ways \(m\) to form a committee with at least 6 males equals the number of ways \(n\) to form a committee with at least 3 females. What is the value of \(m = n\)?</p>
Step-by-Step Solution
Key Concept: When forming committees from two groups, 'at least 6 males from 8' and 'at least 3 females from 5' are complementary selections that must be equal due to symmetry constraints. Recognize that selecting 6+ males automatically determines a specific female count, and this creates a bijection with the 3+ females condition.
<p><strong>Step 1:</strong> Set up m (at least 6 males from 8, total 11 persons).</p><p>If we select k males, we select (11-k) females. For k ≥ 6:</p><p>m = C(8,6)·C(5,5) + C(8,7)·C(5,4) + C(8,8)·C(5,3)</p><p>m = 28·1 + 8·5 + 1·10 = 28 + 40 + 10 = 78</p><p><strong>Step 2:</strong> Set up n (at least 3 females from 5, total 11 persons).</p><p>If we select j females, we select (11-j) males. For j ≥ 3:</p><p>n = C(5,3)·C(8,8) + C(5,4)·C(8,7) + C(5,5)·C(8,6)</p><p>n = 10·1 + 5·8 + 1·28 = 10 + 40 + 28 = 78</p><p><strong>Step 3:</strong> Observe the symmetry: the terms match perfectly because selecting k males from 8 and (11-k) females from 5 with k ∈ {6,7,8} gives the same combinations as selecting j females from 5 and (11-j) males from 8 with j ∈ {3,4,5}.</p><p>∴ m = n = <strong>78</strong></p>
Correct Answer: C