The number of ways of arranging 8 men and 5 women around a circular table such that no two women can sit together is

Options

A. `8!`
B. `4!`
C. `8!4!`
D. `7! xx 8_(P_(5))`

Answer

To solve the problem of arranging 8 men and 5 women around a circular table such that no two women sit together, we can follow these steps:

### Step 1: Arrange the Men
Since we are arranging people around a circular table, we can fix one man to eliminate the effect of rotations. This means we can arrange the remaining 7 men in a linear fashion. The number of ways to arrange 8 men in a circular manner is given by:

\[ (8 - 1)! = 7! \]

### Step 2: Identify Places for Women
Once the men are arranged, we need to determine the available spaces for the women. In a circular arrangement of 8 men, there are 8 gaps (spaces) between them where the women can sit. These gaps are:

- Between man 1 and man 2
- Between man 2 and man 3
- Between man 3 and man 4
- Between man 4 and man 5
- Between man 5 and man 6
- Between man 6 and man 7
- Between man 7 and man 8
- Between man 8 and man 1

### Step 3: Choose Places for Women
We need to select 5 out of these 8 available gaps to place the women. The number of ways to choose 5 gaps from 8 is given by the combination:

\[ \binom{8}{5} \]

### Step 4: Arrange the Women
After selecting the gaps, we can arrange the 5 women in those chosen gaps. The number of ways to arrange 5 women is given by:

\[ 5! \]

### Step 5: Calculate the Total Arrangements
Now, we can calculate the total number of arrangements by multiplying the number of ways to arrange the men, the number of ways to choose the gaps for the women, and the number of ways to arrange the women:

\[ \text{Total arrangements} = 7! \times \binom{8}{5} \times 5! \]

### Step 6: Simplify the Expression
We can simplify \(\binom{8}{5}\) as follows:

\[ \binom{8}{5} = \binom{8}{3} = \frac{8!}{5! \cdot 3!} \]

Thus, the total arrangements can be expressed as:

\[ \text{Total arrangements} = 7! \times \frac{8!}{5! \cdot 3!} \times 5! \]

The \(5!\) cancels out, leading to:

\[ \text{Total arrangements} = 7! \times \frac{8!}{3!} \]

### Final Expression
This can also be expressed as:

\[ \text{Total arrangements} = 7! \times 8P5 \]

Where \(8P5\) represents the number of permutations of 8 items taken 5 at a time.

### Conclusion
Thus, the total number of ways to arrange 8 men and 5 women around a circular table such that no two women sit together is:

\[ 7! \times 8P5 \]