Question

The number of ways can 7 boys be seated at a round table so that 2 particular boys are next to each other is:-
A.6!2!
B.6!
C.5!2!
D.5!

Answer 1

Hint: First, let two boys that sit together be a single entity. So, now arrange 6 entities in 6 places on a circular table. If $n$ objects are arranged in $n$ places in a circular table, then there are $\left( {n - 1} \right)!$ ways. Then, find the number of ways in which the boys that were fixed together can occupy places. Next, multiply both the resultant number of ways to get the required answer.

Complete step-by-step answer:
We want to find the numbers of ways in which 7 boys can be seated on a round table when 2 particular boys are next to each other.
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 $\left( {n - 1} \right)!$ ways.
That is, if there are 6 boys and 6 places, then there are $\left( {6 - 1} \right)! = 5!$ 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!.
Hence, option C is correct.

Note: 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 $\left( {n - 1} \right)!$ ways.