Find the number of ways in which 10 persons can sit round a circular table so that none of them has the same neighbours in any two arrangements.

Answer

To solve the problem of finding the number of ways in which 10 persons can sit around a circular table such that none of them has the same neighbors in any two arrangements, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Circular Arrangements**:
- When arranging people in a circle, we fix one person to eliminate the effect of rotations. This means we can arrange the remaining people around this fixed person.

2. **Fixing One Person**:
- If we fix one person, we have 9 remaining persons to arrange. The number of ways to arrange these 9 persons in a line is given by \(9!\) (factorial of 9).

3. **Considering Clockwise and Anticlockwise Arrangements**:
- Arrangements in a circle can be viewed in two ways: clockwise and anticlockwise. However, these two arrangements are considered the same in circular permutations. Therefore, we need to divide our total arrangements by 2 to account for this.

4. **Calculating the Total Arrangements**:
- The total number of arrangements of 10 persons around a circular table is given by:
\[ \text{Total arrangements} = \frac{(n-1)!}{2} = \frac{9!}{2} \]

5. **Calculating \(9!\)**:
- Now, we calculate \(9!\):
\[ 9! = 362880 \]

6. **Final Calculation**:
- Now, we divide \(9!\) by 2:
\[ \text{Total arrangements} = \frac{362880}{2} = 181440 \]

Thus, the number of ways in which 10 persons can sit around a circular table such that none of them has the same neighbors in any two arrangements is **181440**.