<p>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 is given by</p>
Step-by-Step Solution
Key Concept: For a round table with the constraint that no two women sit together, first arrange the men in a circle (fixing one position to account for rotational symmetry), then place women in the gaps created between men.
<p><strong>Step 1:</strong> Arrange 6 men around a circular table. For circular arrangements, fix one position to eliminate rotational counting: <strong>(6-1)! = 5! = 120 ways</strong></p><p><strong>Step 2:</strong> When 6 men sit in a circle, they create exactly 6 gaps (spaces between consecutive men) where women must sit to ensure no two women are adjacent.</p><p><strong>Step 3:</strong> We have 5 women to place in 6 available gaps. This is a permutation problem: choose 5 gaps from 6 and arrange 5 women in them: <strong>P(6,5) = 6!/(6-5)! = 6!/1! = 720 ways</strong></p><p><strong>Step 4:</strong> By the multiplication principle, total arrangements = 5! × P(6,5) = <strong>120 × 720 = 86,400</strong></p><p>∴ Answer: A</p>
Correct Answer: A