SOLUTION: In how many ways can 7 people be seated at a round table if 2 particular people must not sit next to each other? I need an explanatory answer. Thank you! :)

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Question 768428: In how many ways can 7 people be seated at a round table if 2 particular people must not sit next to each other?
I need an explanatory answer.
Thank you! :)
Found 2 solutions by Edwin McCravy, abdullahkhawer:
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
In how many ways can 7 people be seated at a round table if 2 particular people must not sit next to each other?
The formula for n people to sit 

1. in a straight line is n!.

2. at a round table there is (n-1)!.

If it would not matter if the 2 particular people sat next to each other
then the answer would be (7-1)! or 6! or 720.

But from those 6! or 720 ways, we must subtract the number of ways
those two particular persons can sit next to each other.

The cases we must subtract are the ways of seating 6 "things" around the
table.  5 single people and and one "pair".  That would be (6-1)! or 5! 
or 120 ways.

However, there are two ways the "pair"  could sit, A left of B or A right
of B.  So we double the 120 ways to 240 ways.  So we must subtract 240 ways
from the 720.

Answer: 720-240 = 480 ways.

Edwin
Answer by abdullahkhawer(5)   (Show Source): You can put this solution on YOUR website!
7 People: Person 1, Person 2, 3,4,5,6, and 7
at a round table
7 Chairs: 1,2,3,4,5,6,7
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 - can sit in 5! 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 - can be rearranged in 5! ways
Therefore, total number of ways in which Person 1 and Person 2 are next to each other is: 2*5! = 120*2 = 240
You then have to find in how many ways they are not sitting next to each other; subtract 240 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 7
Arrangement = (7-1)! = 6!= 720
Number of ways Person 1 and Person 2 are not next to each other = 720-240 = 480
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