<p>A committee of 11 members is to be formed from 8 males and 5 females. If <em>m</em> is the number of ways the committee is formed with at least 6 males and <em>n</em> is the number of ways the committee is formed with at least 3 females, then:</p>
Step-by-Step Solution
Key Concept: Break the problem into mutually exclusive cases based on the gender constraint, then count each case separately. Use the constraint that total committee size = 11 to determine possible distributions of males and females.
<p><strong>Step 1: Find m (at least 6 males, so at most 5 females)</strong></p><p>With 11 members total from 8 males and 5 females:</p><ul><li>6 males, 5 females: C(8,6) × C(5,5) = 28 × 1 = 28</li><li>7 males, 4 females: C(8,7) × C(5,4) = 8 × 5 = 40</li><li>8 males, 3 females: C(8,8) × C(5,3) = 1 × 10 = 10</li></ul><p><strong>m = 28 + 40 + 10 = 78</strong></p><p><strong>Step 2: Find n (at least 3 females, so at most 8 males)</strong></p><ul><li>3 females, 8 males: C(5,3) × C(8,8) = 10 × 1 = 10</li><li>4 females, 7 males: C(5,4) × C(8,7) = 5 × 8 = 40</li><li>5 females, 6 males: C(5,5) × C(8,6) = 1 × 28 = 28</li></ul><p><strong>n = 10 + 40 + 28 = 78</strong></p><p><strong>Step 3: Analyze relationship</strong></p><p>Notice that m = n = 78 (the cases are complementary distributions). This makes sense because 'at least 6 males' is equivalent to 'at most 5 females,' which is the complement of 'at least 3 females' when selecting exactly 11 people.</p><p>∴ <strong>Answer: D</strong> (m = n)</p>
Correct Answer: D