The number of ways can 7 boys be seated at a round table so that 2 particular boys are separated is :

The correct answer is:

 Step by step solution :
We want to find the numbers of ways in which 7 boys can be seated on a round table when 2 particular boys are seperated.
Let us assume that two boys put together a single unit.
Now, we will arrange 6 entities in 6 places on a circular table.
When n objects are arranged in n places in a circular table, then there are (n−1)! ways.
That is, if there are 6 boys and 6 places, then there are (6−1)!  ways to arrange.
Now, the two boys that were fixed in a single place can also rearrange themselves.
The number of ways in which two boys can arrange among themselves is 2!
Next, to find the number of ways 7 boys can be seated at a round table so that 2 particular boys are next to each other is multiply 5! with 2!
That is, total ways possible are 5!2!.
Total ways of arranging 2 boys sitting together = 5! 2! = 240.
Total ways of arranging 7 boys at a round table = (7–1)! = 6! = 720

Total number of ways in which they are separated  = Total ways of arranging 7 boys at a round table - Total ways of arranging 2 boys sitting together
Thus, Total number of ways in which they are separated  = 720 - 240 = 480

When n distinct objects are arranged in n places, then the total number of ways possible are n!. But, when n objects are to arranged in n places on a circular table, there are (n−1)!ways.