Question

The number of ways of arranging 9 men and 5 women around a circular table so that no two women come together are:

• A. \( 8! \, 8P5 \)
• B. \( 9! \, 9P5 \)
• C. \( 8! \, 9P5 \)
• D. \( 8! \, 5! \)

Hint:

In circular arrangements, always fix one element in place to avoid identical rotations. For seating problems like this, focus on the number of available spaces and how to select them for the other elements.

Answer 1

The Correct Option is C

Step 1: First, arrange the 9 men around the circular table. Since the arrangement is circular, the number of ways to arrange the men is \( (9 - 1)! = 8! \).
Step 2: Now, for the women to be seated such that no two women sit together, there must be a woman seated between every two men. There are 9 spaces created by the men for the women to sit. We need to choose 5 positions out of these 9 spaces for the women.
The number of ways to select 5 spaces from the 9 available spaces is \( 9P5 \).
Step 3: For each chosen space, there are 5! ways to arrange the women in those spaces.
Thus, the total number of ways to arrange the 9 men and 5 women so that no two women sit together is: \[ 8! \times 9P5 \]