In how many ways can 10 people sit around a table so that all shall not have the same neighbours in any two arrangement ?

Answer

To solve the problem of how many ways 10 people can sit around a table such that no two arrangements have the same neighbors, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Circular Permutations:**
When arranging people in a circle, we fix one person to eliminate the effect of rotations. This means that the arrangement of the remaining people can be treated as a linear arrangement.

2. **Calculating Arrangements:**
For 10 people, if we fix one person, we have 9 remaining people to arrange. The number of ways to arrange these 9 people is given by \( (n-1)! \), where \( n \) is the total number of people. Therefore, we have:
\[ \text{Arrangements} = (10 - 1)! = 9! \]

3. **Considering Clockwise and Anti-clockwise Arrangements:**
Since arrangements that are mirror images (clockwise vs anti-clockwise) are considered the same, we need to divide the total arrangements by 2. Thus, the number of distinct arrangements is:
\[ \text{Distinct Arrangements} = \frac{9!}{2} \]

4. **Calculating Factorial:**
Now, we calculate \( 9! \):
\[ 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362880 \]

5. **Final Calculation:**
Now, we divide \( 9! \) by 2 to find the number of distinct arrangements:
\[ \text{Distinct Arrangements} = \frac{362880}{2} = 181440 \]

### Final Answer:
Thus, the total number of ways 10 people can sit around a table such that no two arrangements have the same neighbors is:
\[ \text{Answer} = 181440 \]