<p>There are 5 eligible Punjabi grooms of which 3 know Bengali and 5 eligible Bengali grooms of which 2 know Punjabi. There are 5 eligible Punjabi brides and 5 eligible Bengali brides. If an eligible groom is agreeable to marry a girl of his community or knowing her language and brides have no choice, in how many different ways 10 couples can be formed?</p>
Step-by-Step Solution
Key Concept: Partition grooms into language-matched groups (Punjabi-only, Bengali-only, bilingual), then match each group with appropriate brides respecting the constraint that grooms marry within community OR can communicate linguistically.
<p><strong>Step 1: Categorize Punjabi grooms</strong></p><p>Punjabi grooms: 5 total → 2 know only Punjabi, 3 know both Punjabi and Bengali</p><p><strong>Step 2: Categorize Bengali grooms</strong></p><p>Bengali grooms: 5 total → 3 know only Bengali, 2 know both Bengali and Punjabi</p><p><strong>Step 3: Determine valid marriages for Punjabi grooms</strong></p><p>• The 2 Punjabi-only grooms must marry Punjabi brides (2 ways to choose × 2! arrangements with remaining Punjabi brides)</p><p>• The 3 bilingual Punjabi grooms can marry either Punjabi or Bengali brides</p><p><strong>Step 4: Determine valid marriages for Bengali grooms</strong></p><p>• The 3 Bengali-only grooms must marry Bengali brides (3 ways)</p><p>• The 2 bilingual Bengali grooms can marry either Bengali or Punjabi brides</p><p><strong>Step 5: Resolve the matching</strong></p><p>Case 1: All bilingual grooms marry within community</p><p>• 2 Punjabi-only + 3 bilingual Punjabi grooms → 5 Punjabi brides: 5! ways</p><p>• 3 Bengali-only + 2 bilingual Bengali grooms → 5 Bengali brides: 5! ways</p><p>• Total: 5! × 5! = 120 × 120 = 14,400 ways</p><p>Case 2: Bilingual grooms marry outside (no valid complete matching exists without violating constraints)</p><p><strong>Step 6: Verify coverage</strong></p><p>The only valid scenario is Case 1, where all groups remain within community.</p><p>∴ Answer: <strong>14,400</strong> or <strong>120 × 120 = (5!)²</strong></p>
Correct Answer: 14