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.

 

[Architeuthis Dux]

I would approach this problem using the principles of combinatorics and probability. The first step would be to determine the total number of possible seating arrangements for 7 people around a table without any restrictions. This can be calculated using the formula n!, where n is the number of people.

In this case, there are 7 people, so the total number of possible arrangements is 7! = 5040.

Next, we need to consider the restriction that A does not want to be next to B. To solve this, we can treat A and B as a single unit, and calculate the number of ways to arrange this unit with the other 5 people. This can be done in (5-1)! = 4! ways.

However, we also need to consider the fact that A and B can be arranged in two different ways (A next to B or B next to A). Therefore, the total number of arrangements where A and B are together is 2 x 4! = 48.

Finally, we can subtract this number from the total number of possible arrangements to get the final answer, which is 5040 - 48 = 4992.

In conclusion, there are 4992 possible seating arrangements for 7 people around a table if A does not want to be next to B.