Permutations & Combinations
Selection with Additional Conditions
Grade 11
Question:
<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>
<p>(a) 380</p>
<p>(b) 320</p>
<p>(c) 260</p>
<p>(d) 95</p>
Step-by-Step Solution
Key Concept: Break into cases based on the constraint 'at most one boy', then select members and separately choose the captain from the selected team.
<p><strong>Solution:</strong> At most one boy means either 0 boys or 1 boy.</p><p><strong>Case 1: 0 boys and 4 girls</strong></p><p>Select 4 girls from 6: $\binom{6}{4} = 15$</p><p>Select captain from these 4: 4 ways</p><p>Total for Case 1: $15 \times 4 = 60$</p><p><strong>Case 2: 1 boy and 3 girls</strong></p><p>Select 1 boy from 4: $\binom{4}{1} = 4$</p><p>Select 3 girls from 6: $\binom{6}{3} = 20$</p><p>Select captain from these 4: 4 ways</p><p>Total for Case 2: $4 \times 20 \times 4 = 320$</p><p>Total ways: $60 + 320 = 380$</p><p>∴ Answer is (a) 380.</p>
Correct Answer: A