<p>Find the number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together.</p>
Step-by-Step Solution
Key Concept: First arrange men in a circle (fixing one position to account for rotational symmetry), then place women in the gaps created between men. With 6 men arranged circularly, exactly 6 spaces are available for women.
<p><strong>Step 1: Arrange men in a circle</strong></p><p>The number of ways to arrange 6 men around a round table is (6-1)! = 5! = 120 (fixing one man's position to eliminate rotational counting).</p><p><strong>Step 2: Identify positions for women</strong></p><p>When 6 men are seated around a circular table, they create exactly 6 gaps (spaces between consecutive men where women can sit).</p><p><strong>Step 3: Place women in gaps</strong></p><p>To ensure no two women sit together, we must place each of the 5 women in different gaps. Choose 5 gaps from 6 available gaps and arrange 5 women in them: P(6,5) = 6!/(6-5)! = 6!/1! = 6! = 720.</p><p>Alternatively: Select 5 gaps from 6 gaps in C(6,5) = 6 ways, then arrange 5 women in 5! ways = 6 × 5! = 6!.</p><p><strong>Step 4: Apply multiplication principle</strong></p><p>Total arrangements = (Arrangements of men) × (Arrangements of women in gaps)</p><p>= 5! × 6!</p><p><strong>∴ Answer: 6! × 5! = 720 × 120 = 86,400</strong></p>
Correct Answer: 6! × 5!