<p>Eighteen guests are to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three others on the other side. Determine the number of different ways in which the sitting arrangement can be made.</p>
Step-by-Step Solution
Key Concept: Partition the constraint satisfaction problem: First seat the guests with fixed side preferences, then fill remaining seats on each side, then arrange all seated guests in their respective rows.
<p><strong>Step 1: Understand the constraint</strong></p><p>Total guests: 18 (9 on each side). Four particular guests want the same side, three others want the opposite side.</p><p><strong>Step 2: Assign guests to sides</strong></p><p>Place 4 guests on Side A (their choice) and 3 guests on Side B (their choice). This uses 4 + 3 = 7 guest slots.</p><p><strong>Step 3: Fill remaining seats</strong></p><p>Remaining guests: 18 - 7 = 11 guests to be distributed: 5 more needed on Side A (to make 9), and 6 more needed on Side B (to make 9).</p><p>Ways to choose 5 from these 11 remaining guests for Side A: C(11,5)</p><p><strong>Step 4: Arrange guests on each side</strong></p><p>The 9 guests on Side A can be arranged in 9! ways along that side.</p><p>The 9 guests on Side B can be arranged in 9! ways along that side.</p><p><strong>Step 5: Calculate total arrangements</strong></p><p>Total = C(11,5) × 9! × 9!</p><p>= 462 × 362,880 × 362,880</p><p>= 462 × (9!)²</p><p><strong>∴ Answer: 462 × (9!)² or equivalently C(11,5) × 9! × 9!</strong></p>
Correct Answer: 462