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Maths Question 16 – AP-EAMCET 2026

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The number of ways of arranging 10 men and 5 women around a circular table such that no two women sit together is

When dealing with arrangements where certain items cannot sit together, first arrange the items that do not have this restriction. These items will create gaps where the restricted items can then be placed.

Step 1: Arranging Men in a Circle✦ Active

First, we arrange the 10 men around the circular table. For a circular permutation of n distinct items, the number of ways is (n−1)!.

Number of ways to arrange 10 men=(10−1)!=9!
💡 Teacher's Secret Hint

Remember that circular permutations fix one position to avoid overcounting rotations. This is why we use (n−1)! instead of n!.

Step 2: Identifying Available Spaces for Women○ Expand

When 10 men are arranged in a circle, they create 10 distinct spaces between them where the women can be seated. For no two women to sit together, each woman must occupy one of these 10 spaces, and no two women can share a space.

Step 3: Placing Women in the Available Spaces○ Expand

We need to choose 5 of the 10 available spaces and arrange the 5 women in them. This is a permutation problem, as the women are distinct and the order in which they are placed in the chosen spaces matters.

Number of ways to place 5 women in 10 spaces=10P5=10!(10−5)!=10!5!
💡 Teacher's Secret Hint

It's a permutation, not a combination, because the women are distinct individuals (W1, W2, etc.), and swapping two women between two chosen seats would result in a different arrangement.

Step 4: Combining Arrangements for Total Ways○ Expand

To find the total number of ways, we multiply the number of ways to arrange the men by the number of ways to place the women.

Total number of ways=(Ways to arrange men)×(Ways to place women) Total number of ways=9!×10P5

This matches option 4.

💡 Teacher's Secret Hint

Always multiply the independent choices to get the total number of arrangements for sequential or dependent events.

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