In how many ways 7 men and 7 women can sit on a round table such that no two women sit together

Options

A. `(7 !)^2`
B. `7 ! XX 6!`
C. `(6!)^2`
D. `7!`

Answer

To solve the problem of how many ways 7 men and 7 women can sit at a round table such that no two women sit together, we can follow these steps:

### Step 1: Arrange the Men
Since we have a round table, we can fix one man in a position to eliminate the effect of rotations. This means we only need to arrange the remaining 6 men. The number of ways to arrange 7 men in a circular manner is given by (n-1)!, where n is the number of men.

\[ \text{Ways to arrange men} = (7 - 1)! = 6! = 720 \]

### Step 2: Identify Positions for Women
Once the men are seated, they will create 7 gaps (or positions) where the women can sit. These gaps are located between each pair of men and at the ends of the row of men. Since we want to ensure that no two women sit together, we can only place one woman in each gap.

### Step 3: Arrange the Women
Now that we have 7 gaps available for the 7 women, we can arrange the women in these gaps. The number of ways to arrange 7 women in the 7 gaps is given by 7!, since each woman can occupy any of the gaps.

\[ \text{Ways to arrange women} = 7! = 5040 \]

### Step 4: Calculate Total Arrangements
To find the total number of arrangements where no two women sit together, we multiply the number of ways to arrange the men by the number of ways to arrange the women.

\[ \text{Total arrangements} = \text{Ways to arrange men} \times \text{Ways to arrange women} = 6! \times 7! = 720 \times 5040 \]

Calculating this gives:

\[ 720 \times 5040 = 3,628,800 \]

### Final Answer
Thus, the total number of ways for 7 men and 7 women to sit at a round table such that no two women sit together is **3,628,800**.

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