<p>Kanchan has 10 friends among whom two are married to each other. She wishes to invite five of them for a party. If the married couples refuse to attend separately, then the number of different ways in which she can invite five friends is</p>
Step-by-Step Solution
Key Concept: Use complementary counting: Total ways to choose 5 from 10 minus the invalid cases where exactly one person from the married couple is invited. Alternatively, partition into cases: both married couple members, neither member, or use direct counting.
<p><strong>Step 1: Identify the constraint</strong> - The married couple (2 people) refuse to attend separately, meaning either both attend together or neither attends.</p><p><strong>Step 2: Partition into valid cases</strong></p><p>• <strong>Case 1:</strong> Both married couple members are invited<br/>Choose the couple (1 way) + choose 3 more from remaining 8 friends = C(8,3) = 56</p><p>• <strong>Case 2:</strong> Neither married couple member is invited<br/>Choose 5 from remaining 8 friends = C(8,5) = 56</p><p><strong>Step 3: Apply addition principle</strong><br/>Total valid ways = C(8,3) + C(8,5) = 56 + 56 = 112</p><p><strong>Verification by complementary method:</strong><br/>Total ways to choose 5 from 10 = C(10,5) = 252<br/>Invalid ways (exactly 1 from married couple) = C(2,1) × C(8,4) = 2 × 70 = 140<br/>Valid ways = 252 - 140 = 112 ✓</p><p>∴ <strong>Answer: 112</strong></p>
Correct Answer: ABC