<p>Three married couples sit in a row. Find the probability that no husband sits with his wife.</p>
Step-by-Step Solution
Key Concept: Use inclusion-exclusion principle to count derangement-like arrangements where no couple sits adjacent. Calculate total arrangements minus arrangements with at least one couple together, accounting for overlaps systematically.
<p><strong>Step 1:</strong> Total arrangements of 3 couples (6 people) in a row = 6! = 720</p><p><strong>Step 2:</strong> Let A_i = event that couple i sits together (i = 1,2,3). We need P(no couple together) = P(A₁' ∩ A₂' ∩ A₃')</p><p><strong>Step 3:</strong> By inclusion-exclusion: |A₁' ∩ A₂' ∩ A₃'| = 720 - |A₁ ∪ A₂ ∪ A₃|</p><p><strong>Step 4:</strong> Calculate |A₁ ∪ A₂ ∪ A₃|:</p><p>• |A₁| = |A₂| = |A₃| = 5! × 2! = 120 × 2 = 240 (treat couple as unit, arrange 5 units, arrange within couple)</p><p>• |A₁ ∩ A₂| = |A₁ ∩ A₃| = |A₂ ∩ A₃| = 4! × 2! × 2! = 24 × 4 = 96 (two couples as units)</p><p>• |A₁ ∩ A₂ ∩ A₃| = 3! × 2! × 2! × 2! = 6 × 8 = 48 (three couples as units)</p><p><strong>Step 5:</strong> |A₁ ∪ A₂ ∪ A₃| = 3(240) - 3(96) + 48 = 720 - 288 + 48 = 480</p><p><strong>Step 6:</strong> Favorable outcomes = 720 - 480 = 240</p><p><strong>Step 7:</strong> Probability = 240/720 = 1/3</p><p>∴ Answer: A (Probability = 1/3)</p>
Correct Answer: A