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  • A group of 8 people is going to sit at a round table. In how many different ways can they be seated if two particular people must always be seated next to each other, and

A group of 8 people is going to sit at a round table. In how many different ways can they be seated if two particular people must always be seated next to each other, and the remaining people can sit in any order?

 

Option: 1

720


Option: 2

640


Option: 3

440


Option: 4

560


Answers (1)

best_answer

To calculate the number of different ways the group of 8 people can be seated at a round table, with two particular people always seated next to each other, and the remaining people able to sit in any order, we can consider the following:

We treat the two particular people who must always be seated next to each other as a single entity. Therefore, we have 7 entities to arrange: the combined entity of the two particular people, and the remaining 6 people.

The number of different ways to arrange these 7 entities in a row is 7 !.

However, when seated at a round table, we need to consider that the arrangement can be rotated without changing the relative positions of the people.

Since there are 7 entities, there are 7 possible starting points for the rotation.

Therefore, we divide the total number of arrangements by 7 to account for the rotations.

Hence, the number of different ways the group of 8 people can be seated at a round table, with two particular people always seated next to each other, and the remaining people able to sit in any order, is:

7 ! / 7=6 !=720 \text {. }

Therefore, there are 720 different ways the group of 8 people can be seated at a round table, satisfying the given conditions.

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avinash.dongre

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