Circle Permutations: 7 People, A Not Next to B

[Cyborg31]

Homework Statement

7 people around a table, how many ways of seating if A does not want to be next to B?

Homework Equations

(n-1)!

The Attempt at a Solution

Well I know the number of ways to get 7 people around a table is 6! but not sure how to solve it if A does not want to be next to B.

 

[HallsofIvy]

Seat A first anywhere at the table. There are 6 people left to seat but one of those, B, cannot be seated next to A. That means there are 5 people who could be seated on A's right side. After that choice is made, there are 4 people who could be seated on A's left side. Once you have people seated on either side of A, you can put B back into the "mix". There are now 4 people to choose the next person to seat from, then 3, then 2, then 1.