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The number of ways in which 10 boys can ...

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

A

`2xx48!`

B

12

C

360

D

`7xx8!`

Text Solution

AI Generated Solution

The correct Answer is:
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-by-Step Solution: 1. **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, one position is fixed to avoid counting rotations as different arrangements. 2. **Arrangements Where Two Boys Sit Together**: To find the arrangements where the two particular boys (let's call them A and B) sit together, we can treat them as a single unit or block. This means we now have 9 units to arrange (the block AB and the other 8 boys). The number of arrangements of these 9 units around the table is: \[ (9 - 1)! = 8! = 40320 \] However, within the block, A and B can be arranged in 2 ways (AB or BA). Therefore, the total arrangements where A and B sit together is: \[ 8! \times 2 = 40320 \times 2 = 80640 \] **Hint**: When treating two people as a single unit, always remember to multiply by the arrangements of the individuals within that unit. 3. **Arrangements Where Two Boys Do Not Sit Together**: To find the arrangements where A and B do not sit next to each other, we subtract the arrangements where they sit together from the total arrangements: \[ \text{Arrangements where A and B do not sit together} = \text{Total arrangements} - \text{Arrangements where A and B sit together} \] \[ = 9! - (8! \times 2) = 362880 - 80640 = 282240 \] **Hint**: This is a common technique in combinatorics known as complementary counting, where you count the total arrangements and subtract the unwanted cases. ### Final Answer: 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: \[ \boxed{282240} \]
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