<p>Find the number of ways of selecting 3 pairs from 8 distinct objects.</p>
Step-by-Step Solution
Key Concept: When selecting pairs from distinct objects, we first choose 2 objects for each pair (using combinations), then divide by 3! because the pairs themselves are indistinguishable (unordered collection of pairs).
<p><strong>Step 1:</strong> Select 2 objects from 8 for the first pair: ⁸C₂ ways</p><p><strong>Step 2:</strong> Select 2 objects from remaining 6 for the second pair: ⁶C₂ ways</p><p><strong>Step 3:</strong> Select 2 objects from remaining 4 for the third pair: ⁴C₂ ways</p><p><strong>Step 4:</strong> Since the three pairs form an unordered collection (pair 1, pair 2, pair 3 are indistinguishable), divide by 3! to eliminate overcounting</p><p><strong>Calculation:</strong> (⁸C₂ × ⁶C₂ × ⁴C₂)/3! = (28 × 15 × 6)/6 = 2520/6 = 420</p><p><strong>∴ Answer: (⁸C₂ × ⁶C₂ × ⁴C₂)/3! = 420</strong></p>
Correct Answer: (⁸C₂ × ⁶C₂ × ⁴C₂)/3!