The number of ways in which 10 boys can take positions around a circular round table, if two particular boys must not be seated side by side, is

Options

A. `2xx48!`
B. 12
C. 360
D. `7xx8!`

Answer

To solve the problem of arranging 10 boys around a circular table such that two particular boys do not sit next to each other, we can follow these steps:

### Step 1: Calculate the total arrangements without restrictions
When arranging \( n \) people around a circular table, the number of arrangements is given by \( (n-1)! \). For 10 boys, the total arrangements without any restrictions is:
\[ (10-1)! = 9! = 362880 \]

**Hint:** Remember that in circular arrangements, we fix one position to avoid counting rotations as different arrangements.

### Step 2: Calculate arrangements where the two particular boys are together
To find the arrangements where the two particular boys (let's call them A and B) are sitting together, we can treat A and B as a single unit or block. This means we now have 9 units to arrange (the block of A and B plus the other 8 boys).

The number of arrangements of these 9 units around the table is:
\[ (9-1)! = 8! = 40320 \]
Within the block, A and B can be arranged in 2 ways (AB or BA). Therefore, the total arrangements where A and B are together is:
\[ 8! \times 2 = 40320 \times 2 = 80640 \]

**Hint:** When treating two people as a single unit, remember to account for their internal arrangements.

### Step 3: Subtract the restricted arrangements from the total arrangements
Now, we can find the number of arrangements where A and B are not sitting together by subtracting the arrangements where they are together from the total arrangements:
\[ 9! - (8! \times 2) = 362880 - 80640 = 282240 \]

**Hint:** This step uses the principle of complementary counting: subtracting the unwanted arrangements from the total arrangements.

### Conclusion
The number of ways in which 10 boys can take positions around a circular round table, ensuring that the two particular boys do not sit side by side, is:
\[ \boxed{282240} \]