6 men and 6 women are to be seated at a circular table such that no two women are adjacent , then the number of arrangements is

Options

A. 86400
B. 46800
C. 12!
D. `1/2 (12!)`

Answer

To solve the problem of seating 6 men and 6 women at a circular table such that no two women are adjacent, we can follow these steps:

### Step 1: Arrange the Men
Since we are dealing with a circular arrangement, we can fix one man's position to eliminate the effect of rotations. This means we only need to arrange the remaining 5 men.

- The number of ways to arrange 6 men in a circular manner is given by (n-1)! where n is the number of men.
- Here, n = 6, so the number of arrangements is (6-1)! = 5! = 120.

### Step 2: Identify the Seats for Women
Once the men are seated, we need to determine where the women can sit. Since no two women can sit next to each other, they must occupy the spaces between the men.

- With 6 men seated, there will be 6 gaps created between them (one gap between each pair of men).

### Step 3: Arrange the Women
Now we need to place the 6 women in these 6 gaps. Since there are exactly 6 gaps and 6 women, each gap will be occupied by one woman.

- The number of ways to arrange the 6 women in the 6 gaps is given by 6! = 720.

### Step 4: Calculate the Total Arrangements
To find the total number of arrangements, we multiply the number of arrangements of men by the number of arrangements of women.

- Total arrangements = (Arrangements of men) × (Arrangements of women) = 5! × 6! = 120 × 720 = 86400.

Thus, the total number of arrangements of seating 6 men and 6 women at a circular table such that no two women are adjacent is **86400**.

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