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

Options

A. 7!
B. `7!xx6!`
C. `(6!)^(2)`
D. `(7!)^(2)`

Answer

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

### Step-by-Step Solution:

1. **Arrange the Men**:
Since we are seating people around a round table, we can fix one man's position to eliminate the circular permutations. The remaining 6 men can then be arranged in the remaining seats. The number of ways to arrange 7 men around a round table is given by:
\[ (n-1)! = (7-1)! = 6! = 720 \]

2. **Identify Spaces for Women**:
After seating the 7 men, there will be 7 gaps created between them where the women can sit. These gaps can be visualized as follows:
- M _ M _ M _ M _ M _ M _ M
Here, "M" represents a man and "_" represents a gap where a woman can sit.

3. **Arrange the Women**:
We need to place the 7 women in the 7 available gaps. Since no two women can sit together, we can place one woman in each gap. The number of ways to arrange 7 women in these 7 gaps is:
\[ 7! = 5040 \]

4. **Calculate the Total Arrangements**:
The total number of arrangements is the product of the arrangements of men and women:
\[ \text{Total arrangements} = 6! \times 7! = 720 \times 5040 \]

5. **Perform the Calculation**:
Now, we calculate:
\[ 720 \times 5040 = 3628800 \]

Thus, the total number of ways to seat 7 men and 7 women around a round table such that no two women sit together is **3,628,800**.