10 persons are to be arranged in a circular fashion so that in no two arrangements all the persons have same neighbours. The number of ways of doing so is equal to:

Options

A. (a) Number of ways of arranging 10 people around a circular table divided by 2
B. (b) Number of different garlands that can be formed using 10 different flowers.
C. (c) Number of different necklaces that can be formed using 10 different beads
D. (d) Number of different garlands that can be formed using 10 identical flowers.

Answer

To solve the problem of arranging 10 persons in a circular fashion such that no two arrangements have the same neighbors, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Circular Arrangements**:
In a circular arrangement, one person can be fixed to eliminate identical rotations. For `n` persons, the number of ways to arrange them in a circle is given by `(n - 1)!`.

**Hint**: Remember that fixing one person helps to avoid counting the same arrangement multiple times due to rotation.

2. **Calculating Arrangements**:
For 10 persons, the number of arrangements in a circular fashion is:
\[ (10 - 1)! = 9! \]

**Hint**: The factorial function `n!` represents the product of all positive integers up to `n`.

3. **Considering Neighbor Restrictions**:
The problem states that no two arrangements should have the same neighbors. This means that if we have an arrangement of persons, swapping two adjacent persons would create an arrangement that is not allowed. To account for this restriction, we need to divide the total arrangements by 2.

**Hint**: Think about how swapping neighbors creates identical arrangements that should not be counted.

4. **Final Calculation**:
Therefore, the total number of arrangements of 10 persons in a circular fashion, ensuring no two arrangements have the same neighbors, is:
\[ \frac{9!}{2} \]

**Hint**: Dividing by 2 accounts for the fact that swapping two adjacent persons results in the same arrangement.

5. **Conclusion**:
The final answer for the number of ways to arrange 10 persons in a circular fashion under the given conditions is:
\[ \frac{9!}{2} \]

### Final Answer:
\[ \frac{9!}{2} = \frac{362880}{2} = 181440 \]