Circular tables always complicate things.

When maths questions talk about round tables they assume no external points of reference. In other words rotations are the same thing.

6^4 would certainly be the number of possible combinations if the people were in a nice straight row but it is too high for a round table.

For instance

1234 is the same as 2341, you cannot count that combination twice....

1234 = 2341 = 3412 = 4123

Maybe 6^4 should be divided by 4 but I am not sure if this is over simplified.

I'll assume that, even though it could easily be wrong. 6^4/4 = 324

Now now if I chose a number and make the others all different, I get 6*5*4*3 /4 = 120 posibilities.

Now I chose a number make the opposite one the same and the other 2 different I get 6*1*5*4/4 =30possibilities

Now if I have 2 pairs of oposite numbers the same I get 6*1*5*1 /2= 15 posibilities

Makes a total of 165 possibilities where no adjacent ones are the same.

So that could be a prob of 165/324

Please do not believe me. I would be extremely surprised if my answer was not total rubbish.

But I do not think that Coolstuff's answer is correct either.