<p>A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is</p>
Step-by-Step Solution
Key Concept: Break the problem into mutually exclusive cases based on the constraint 'at most one boy' (0 boys or 1 boy), then for each case multiply the ways to select members by the ways to select a captain from those members.
<p><strong>Step 1:</strong> Identify cases based on 'at most one boy' constraint:</p><ul><li><strong>Case 1:</strong> 0 boys, 4 girls</li><li><strong>Case 2:</strong> 1 boy, 3 girls</li></ul><p><strong>Step 2:</strong> For Case 1 (4 girls, 0 boys):</p><ul><li>Select 4 girls from 6: C(6,4) = 15</li><li>Select captain from these 4 girls: 4 ways</li><li>Subtotal: 15 × 4 = 60</li></ul><p><strong>Step 3:</strong> For Case 2 (3 girls, 1 boy):</p><ul><li>Select 3 girls from 6: C(6,3) = 20</li><li>Select 1 boy from 4: C(4,1) = 4</li><li>Total teams: 20 × 4 = 80</li><li>Select captain from these 4 members: 4 ways</li><li>Subtotal: 80 × 4 = 320</li></ul><p><strong>Step 4:</strong> Add both cases (mutually exclusive):</p><p>Total = 60 + 320 = <strong>380</strong></p><p>∴ Answer: A</p>
Correct Answer: A