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