Permutations & Combinations
Team Formation with Restrictions
Grade None
Question:
<p>Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that can be formed from this class, if there are two specific boys A and B, who refuse to be the members of the same team, is ______.</p>
Step-by-Step Solution
Key Concept: Use complementary counting: subtract the 'bad' teams (where both A and B are selected) from the total teams possible. This avoids complex case-by-case analysis.
<p><strong>Step 1:</strong> Find total teams with 2 girls and 3 boys (no restrictions).</p><p>Total = C(5,2) × C(7,3) = 10 × 35 = 350</p><p><strong>Step 2:</strong> Find teams where BOTH A and B are selected (the forbidden case).</p><p>If A and B are both in the team, we need 2 girls and 1 more boy (from remaining 5 boys).</p><p>Bad teams = C(5,2) × C(5,1) = 10 × 5 = 50</p><p><strong>Step 3:</strong> Apply complementary counting.</p><p>Valid teams = Total teams - Bad teams = 350 - 50 = <strong>300</strong></p><p>∴ Answer: <strong>300</strong></p>
Correct Answer: 300