Twelve persons are to be arranged on a round table. IF particular husband and wife refuse to sit side by side then the total no. of arrangments is

Options

A. 9(10!)
B. 2(10!)
C. 45(8!)
D. 10!

Answer

To solve the problem of arranging 12 persons at a round table where a particular husband and wife refuse to sit next to each other, we can follow these steps:

### Step 1: Calculate Total Arrangements Without Restrictions
In a round table arrangement, we can fix one person to eliminate the effect of rotations. Thus, for 12 persons, the total arrangements without any restrictions would be:
\[ (12 - 1)! = 11! \]

### Step 2: Calculate Arrangements Where the Husband and Wife Sit Together
To find the arrangements where the husband (let's call him A) and wife (let's call her B) sit together, we can treat A and B as a single unit or block. This means we now have 11 units to arrange (the A-B block and the other 10 persons).

The arrangements of these 11 units in a round table is:
\[ (11 - 1)! = 10! \]
However, within the block, A and B can switch places, so we multiply by 2:
\[ 2 \times 10! \]

### Step 3: Calculate Arrangements Where the Husband and Wife Do Not Sit Together
To find the arrangements where A and B do not sit together, we subtract the arrangements where they sit together from the total arrangements:
\[ \text{Arrangements where A and B do not sit together} = 11! - 2 \times 10! \]

### Step 4: Simplify the Expression
Now, we can simplify the expression:
\[ 11! = 11 \times 10! \]
Thus,
\[ \text{Arrangements where A and B do not sit together} = 11 \times 10! - 2 \times 10! = (11 - 2) \times 10! = 9 \times 10! \]

### Final Answer
Therefore, the total number of arrangements where the husband and wife do not sit next to each other is:
\[ 9 \times 10! \]