Four men and six women are to be seated along a round table. The number of cases where every two men are separated by to women is

Options

A. 3420
B. 43200
C. 1710
D. None of these

Answer

To solve the problem of seating four men and six women around a round table such that every two men are separated by at least two women, we can follow these steps:

### Step-by-step Solution:

1. **Arrange the Women**:
Since we are seating people around a round table, we can fix one woman to eliminate the effect of rotations. The remaining five women can be arranged in the remaining five positions.
\[ \text{Ways to arrange 6 women} = (6 - 1)! = 5! = 120 \]

**Hint**: Remember that in circular arrangements, fixing one position helps to avoid counting identical arrangements due to rotation.

2. **Identify Positions for Men**:
After seating the six women, we will have six gaps (positions) available for the men (one gap between each pair of women). The arrangement looks like this:
\[ W_1 \_ W_2 \_ W_3 \_ W_4 \_ W_5 \_ W_6 \_ \]
Here, each underscore represents a position where a man can sit.

3. **Choose Positions for Men**:
We need to select 4 out of these 6 available positions for the men. This can be done using combinations:
\[ \text{Ways to choose 4 positions from 6} = \binom{6}{4} = 15 \]

**Hint**: Use the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \) to calculate the number of ways to choose positions.

4. **Arrange the Men**:
Once we have selected the positions for the men, we can arrange the 4 men in these selected positions. The number of ways to arrange 4 men is given by:
\[ \text{Ways to arrange 4 men} = 4! = 24 \]

**Hint**: The factorial of the number of items gives you the total arrangements of those items.

5. **Calculate Total Arrangements**:
Now, we can find the total number of arrangements by multiplying the number of ways to arrange the women, the number of ways to choose positions for the men, and the number of ways to arrange the men:
\[ \text{Total arrangements} = 5! \times \binom{6}{4} \times 4! = 120 \times 15 \times 24 \]

6. **Final Calculation**:
Now, we compute:
\[ 120 \times 15 = 1800 \]
\[ 1800 \times 24 = 43200 \]

Thus, the total number of cases where every two men are separated by at least two women is **43200**.

### Final Answer:
The total number of arrangements is **43200**.