Question

In how many ways could 12 people be

Answer 1

Step-by-step explanation:

There are total 12 People

Person 1, Person 2,3,4,5,6,7, 9, 10, 11 and 12 at a round table

12 Chairs: 1,2,3,4,5,6,7,8,9,10,11 and 12                                                                                

Let chairs 1 and 2 be next to each other

Person 1 is at chair 1 and Person 2 is at chair 2

The rest of the people on 3,4,5,6,7,8,9,10,11 and 12 - can sit in 10! ways

Next, Person 1 and Person 2 change their positions;

Person 1 is sitting at 2 and Person 2 is sitting at 1

The rest of the people on 3,4,5,6,7,8,9,10,11 and 12 - can be rearranged in 10! ways

Therefore, total number of ways in which Person 1 and Person 2 are next to each other is: 2*10! = 7257600

As we then have to find in how many ways they are not sitting next to each other; subtract 7257600 from the total number of possible arrangements.

In circular permutation,

Total number of arrangements of n people = (n-1)!

Here the number of people is 12

Arrangement = (12-1)! = 11!= 39916800

Therefore, the number of ways Person 1 and Person 2 are not next to each other = 39916800 - 7257600 = 2659200

Keywords: permutation, combination

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