<p>In how many ways can 15 members of a council sit along a circular table, when the secretary is to sit on one side of the chairman and the deputy secretary on the other side?</p>
Step-by-Step Solution
Key Concept: Fix the chairman's position to account for circular arrangement (eliminating rotational equivalence), then treat the secretary and deputy secretary as distinguishable ordered pairs on either side of the chairman, leaving 12 remaining members to arrange in the remaining seats.
<p><strong>Step 1:</strong> In circular arrangements, fix one person's position to eliminate rotational counting. Fix the chairman at a specific seat.</p><p><strong>Step 2:</strong> The secretary must sit on one side of the chairman (left or right) and the deputy secretary on the other side. Choose which side each sits: 2 ways (secretary-left & deputy-right OR deputy-left & secretary-right).</p><p><strong>Step 3:</strong> After placing chairman, secretary, and deputy secretary, we have 12 remaining members to arrange in the remaining 12 seats: 12! ways.</p><p><strong>Step 4:</strong> Total arrangements = 2 × 12!</p><p>∴ Answer: <strong>12! × 2</strong></p>
Correct Answer: 12! × 2