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
<em>Keywords: </em><em>permutation, combination</em>
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