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  • A group of 10 people needs to be seated in a row of 10 chairs. If two specific people refuse to sit next to each other, in how many different ways can they be arranged?</

A group of 10 people needs to be seated in a row of 10 chairs. If two specific people refuse to sit next to each other, in how many different ways can they be arranged?

 

Option: 1

125,895

 


Option: 2

856,253

 


Option: 3

445,350

 


Option: 4

685,440


Answers (1)

To calculate the number of different ways the group of 10 people can be seated in a row of 10 chairs, with two specific people not sitting next to each other, we can consider the arrangement in two cases:

Case 1: The two specific people are seated together.

In this case, we treat the two specific people as a single entity. Therefore, we have 9 entities to arrange: the combined entity of the two specific people and the remaining 8 individuals.

The number of different ways to arrange these 9 entities in a row is 9 !.

Case 2: The two specific people are not seated together.

In this case, we need to consider the arrangement of the two specific people with at least one person between them.

We can treat the two specific people as a single entity, and there are 8 possible positions where this entity can be placed with at least one person between them.

Once we have placed this entity, we have 8 entities remaining (including the remaining 8 individuals) to arrange. The number of different ways to arrange these 8 entities in a row is 8 !.

Therefore, the total number of different ways the group of 10 people can be seated in a row of 10 chairs, with the two specific people not sitting next to each other, is:

 

9 !+8 ! \otimes 8=362,880+40,320 \times 8=362,880+322,560=685,440.

Therefore, there are 685,440 different ways the group of 10 people can be arranged in a row of 10 chairs, with the two specific people not sitting next to each other.

Posted by

Ramraj Saini

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