Find the number of ways in which 6 men and 5 women can dine around a circular table if no two women are to sit together.

Answer

To solve the problem of arranging 6 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 dealing with a circular arrangement, we can fix one man to eliminate the effect of rotations. This means we will arrange the remaining 5 men around the table.

- The number of ways to arrange 6 men in a circular manner is given by (n-1)! where n is the number of men.
- Therefore, the number of arrangements for the men is:
\[ (6-1)! = 5! = 120 \]

### Step 2: Identify Spaces for Women
Once the men are arranged, they create spaces where the women can sit. With 6 men seated, there will be 6 gaps (spaces) created between them for the women to occupy.

### Step 3: Place the Women
Since no two women can sit together, we will choose 5 out of the 6 available gaps to place the women. The number of ways to choose 5 gaps from 6 is given by:
\[ \binom{6}{5} = 6 \]

### Step 4: Arrange the Women in the Chosen Gaps
Once we have chosen the gaps, we can arrange the 5 women in those selected gaps. The number of ways to arrange 5 women is:
\[ 5! = 120 \]

### Step 5: Calculate the Total Arrangements
Now, we can find the total number of arrangements by multiplying the number of arrangements of men, the number of ways to choose gaps, and the arrangements of women:
\[ \text{Total arrangements} = \text{Arrangements of men} \times \text{Ways to choose gaps} \times \text{Arrangements of women} \]
\[ = 5! \times \binom{6}{5} \times 5! = 120 \times 6 \times 120 \]

### Step 6: Perform the Final Calculation
Calculating the total:
\[ 120 \times 6 = 720 \]
\[ 720 \times 120 = 86400 \]

Thus, the total number of ways in which 6 men and 5 women can dine around a circular table with no two women sitting together is:
\[ \boxed{86400} \]