12 persons are to be arranged to a round table. If two particular persons among them are not to be side by side, the total number of arrangements, 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 such that two particular persons are not sitting side by side, we can follow these steps:

### Step 1: Calculate the total arrangements without restrictions
When arranging \( n \) persons around a round table, the number of arrangements is given by \( (n-1)! \). For 12 persons, the total arrangements are:
\[ 11! = 39916800 \]

### Step 2: Calculate the arrangements where the two particular persons are sitting together
To find the arrangements where the two specific persons (let's call them A and B) are sitting next to each other, we can treat A and B as a single unit or block. This means we now have 11 units to arrange (the block AB and the other 10 persons).

The number of arrangements for these 11 units around a round table is:
\[ (11-1)! = 10! = 3628800 \]

Since A and B can be arranged within their block in 2 ways (AB or BA), we multiply the arrangements by 2:
\[ 10! \times 2 = 3628800 \times 2 = 7257600 \]

### Step 3: Subtract the arrangements where A and B are together from the total arrangements
Now, we subtract the arrangements where A and B are sitting together from the total arrangements:
\[ 11! - (10! \times 2) = 39916800 - 7257600 = 32659200 \]

### Conclusion
Thus, the total number of arrangements of 12 persons at a round table where two particular persons are not sitting side by side is:
\[ \boxed{32659200} \]

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