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  • A group of 6 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?<st

A group of 6 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?

Option: 1

30


Option: 2

50


Option: 3

60


Option: 4

80


Answers (1)

best_answer

To calculate the number of different ways the group of 6 people can be seated at a round table, with two particular people always seated next to each other, we can consider those two people as a single entity. This reduces the problem to arranging 5 entities ( 4 individuals +1 pair of people) around a circular table.

Now, we can treat the pair of people as a single entity, which means we have 5+1=6 entities to arrange. The number of different arrangements of these entities is given by 6 !.

However, within these arrangements, the two people in the pair can also swap places among themselves, which means we need to divide the total number of arrangements by 2 !.

Additionally, since the people are seated at a round table, we need to account for the rotational symmetry. Since any arrangement can be rotated to create an equivalent arrangement, we divide the total number of arrangements by 6 .

Therefore, the number of different ways the group of 6 people can be seated at a round table, with the two particular people always seated next to each other, is:

6 ! /(2 ! \times 6)=720 /(2 \times 6)=60.
Therefore, there are 60 different ways the group of 6 people can be seated at a round table with the two particular people always seated next to each other.

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seema garhwal

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