SOLUTION: How many ways can 6 students arrange themselves around a circular table if two people must not sit next to each other?

Question 1117917: How many ways can 6 students arrange themselves around a circular table if two people must
not sit next to each other?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

For the case of a circular table, seat the first person in a particular spot and consider the arrangements of the other people. With the first person in a fixed spot, we don't have to consider arrangements that are different only by rotation.

In your problem, first seat one of the two people who can't sit together in a fixed spot.

Then the total number of arrangements without restriction is 5! = 120.

For the restriction, there are 2 spots where the person who can't sit next to the first can NOT sit -- on either side of the first person.

And for each of those 2 choices, the remaining 4 people can sit in any of 4!=24 ways.

So the total number of arrangements that are NOT allowed because of the restriction is 2*24 = 48.

Then the number of possible arrangements with the restriction is 120-48 = 72.>
------------------------------------------------------------

Thanks to tutor ikleyn for providing a solution by a different approach.

I myself am not confident in the way I solve many combinatorics problems; I am always happy to be able to get the same answer to a combinatorics problem by solving it in more than one way.

For a student just learning combinatorics, it is useful to understand different approaches to solving a problem. In a particular problem, you might run into difficulties with one approach but are able to find the solution with another.

Or perhaps you will be like me and just feel better about the work you have done on the problem if you can get the same answer by two different methods.
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
For slightly different formulation and other solution to the same problem see the link

https://www.algebra.com/algebra/homework/Permutations/Permutations.faq.question.1114142.html

https://www.algebra.com/algebra/homework/Permutations/Permutations.faq.question.1114142.html

and also the lesson
    - Persons sitting around a circular table, Problem 4
in this site.

RELATED QUESTIONS
How many ways can 11 people be seated around a circular table if two of them insist to... (answered by ikleyn)
In how many ways can 5 students be seated around a circular table if two specific... (answered by ikleyn,Edwin McCravy)
how many ways can 8 people be seated at a round table if two people must not sit next to... (answered by Edwin McCravy)
In how many ways can 4 boys and 2 girls seat themselves in a row if the two girls are NOT (answered by sudhanshu_kmr)
In how many ways can 4 married couples seat themselves around a circular table if they... (answered by ikleyn)
In how many ways can 7 people be seated at a round table if they can sit anywhere and 2... (answered by Jc0110)
In how many ways can 6 people be arranged to sit around a circular table with 10... (answered by ikleyn)
In how many ways can 7 people be seated at a round table if 2 particular people must not... (answered by Edwin McCravy,abdullahkhawer)
in how many ways can 8 people be seated in a row such that two brothers must not sit next (answered by josmiceli)