Question

There are $10$ distinct chairs around a circular table. The number of ways in which three persons can sit, so that no two consecutive chairs are occupied, is :

• A. $45$
• B. $135$
• C. $300$
• D. None of the above

Answer 1

The correct option is C $300$

The 10 chairs are distinct. The first person can be seated in 10 ways.
The second person cannot be seated on a chair next to the first person. For selecting the chair of the second person, let us consider two cases -
Case I - There is exactly one chair between the first person and the second person.
Hence, the second person can be seated in two ways (either on the left side of the first person or the right side of the first person).
For the third person, two chairs are already occupied and three other chairs are inaccessible, as it would mean sitting adjacent to either the first person or the second person.
Hence, the third person can be seated in 5 ways.
Number of ways $=10\times 2\times 5=100$.
Case II - There are at least 2 chairs between the first and the second person.
The second person can be seated in 5 ways. ( He cannot sit on the chair of the first person as well as the two chairs next to the first person on both sides).
The third person now only has 4 chairs where he can sit. ( 2 chairs are already occupied and two chairs on either side of the two sitting persons are not allowed).
Total ways $=10\times 5\times 4=200$.
Hence, total number of ways $=200+100=300$.