The number of ways in which 7 men and 6 women can dine at a round table, if no two women are to sit together, is given by:

Options

A. `6!xx7!`
B. `42`
C. `5!xx6!`
D. `7!xx5!`

Answer

To solve the problem of arranging 7 men and 6 women at a round table such that no two women sit together, we can follow these steps:

### Step-by-Step Solution:

1. **Fix One Man**:
Since we are arranging people around a round table, we can fix one man in one position to eliminate the effect of rotations. This leaves us with 6 men to arrange.

2. **Arrange the Men**:
The 6 remaining men can be arranged in the remaining 6 positions. The number of ways to arrange these 6 men is given by:
\[ 6! = 720 \]

3. **Identify Positions for Women**:
After arranging the men, there will be 7 gaps created around them where the women can sit (one gap before each man and one after the last man). We can visualize this as:
- M1 _ M2 _ M3 _ M4 _ M5 _ M6 _ M7 _
Here, each underscore represents a potential position for a woman.

4. **Choose Positions for Women**:
We need to choose 6 out of these 7 available gaps to place the women. The number of ways to choose 6 gaps from 7 is given by:
\[ \binom{7}{6} = 7 \]

5. **Arrange the Women**:
Once the positions are chosen, the 6 women can be arranged in those chosen gaps. The number of ways to arrange 6 women is:
\[ 6! = 720 \]

6. **Calculate Total Arrangements**:
The total number of arrangements is the product of the arrangements of men, the choices of gaps, and the arrangements of women:
\[ \text{Total arrangements} = (6!) \times \binom{7}{6} \times (6!) \]
Substituting the values we calculated:
\[ \text{Total arrangements} = 720 \times 7 \times 720 \]

7. **Final Calculation**:
Now, we calculate the total:
\[ 720 \times 7 = 5040 \]
\[ 5040 \times 720 = 3628800 \]

Thus, the total number of ways in which 7 men and 6 women can dine at a round table, ensuring that no two women sit together, is:
\[ \text{Total arrangements} = 720 \times 7 \times 720 = 3628800 \]

### Final Answer:
The total number of arrangements is \( 720 \times 7 \times 720 = 3628800 \).