10 persons are to be seated around a round table so that any two arrangements do not have same neighbourers. The number of ways is given by

Options

A. 9!
B. `1/2 (10) !`
C. `1/2 (9)!`
D. none of these

Answer

To solve the problem of seating 10 persons around a round table such that no two arrangements have the same neighbors, we can follow these steps:

### Step 1: Understanding the Arrangement Around a Round Table
When arranging \( N \) persons around a round table, we can fix one person to eliminate the effect of rotations. This means we only need to arrange the remaining \( N-1 \) persons.

### Step 2: Calculate the Factorial
For \( N = 10 \), we fix one person and arrange the remaining \( 9 \) persons. The number of arrangements is given by:
\[ (10 - 1)! = 9! \]

### Step 3: Considering Neighbor Restrictions
The problem states that no two arrangements should have the same neighbors. This means that if we have an arrangement, we cannot have another arrangement that has the same two neighbors for any person.

### Step 4: Adjusting for Neighbor Restrictions
Since each arrangement can be mirrored (clockwise and counterclockwise), we need to divide the total arrangements by 2 to account for this symmetry. Thus, the number of valid arrangements becomes:
\[ \frac{9!}{2} \]

### Step 5: Calculate the Final Answer
Now we calculate \( 9! \):
\[ 9! = 362880 \]
Then we divide by 2:
\[ \frac{362880}{2} = 181440 \]

### Final Answer
The number of ways to arrange 10 persons around a round table such that no two arrangements have the same neighbors is:
\[ \boxed{181440} \]

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