Question 1117917
<br>
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.<br>
In your problem, first seat one of the two people who can't sit together in a fixed spot.<br>
Then the total number of arrangements without restriction is 5! = 120.<br>
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.<br>
And for each of those 2 choices, the remaining 4 people can sit in any of 4!=24 ways.<br>
So the total number of arrangements that are NOT allowed because of the restriction is 2*24 = 48.<br>
Then the number of possible arrangements with the restriction is 120-48 = 72.>
------------------------------------------------------------<br>
Thanks to tutor ikleyn for providing a solution by a different approach.<br>
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.<br>
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.<br>
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.