seating at a circular table

[Clifford]

In how many ways can 4 people be seated around a circular table with 4 identical chairs?

From a note I took in a previous class, I have a note saying that it seems like the same question as them sitting in a row (which would be 4!) but the order starting at chair 1 is the same as chair 2 and so on. This is true all the way around the table.

so would it not be 4! / 4 = 3! = 6?

 

[pka]

Arranging n different objects in a closed ring can be done in (n-1)! ways.
That is (n!)/n to account for the n identical rotations of the ring.

 

[Tuugii]

that's right! P(3,3)=3!=6.

its really 4x3x2x1 ways of arranging them. but seating in a circle is about how do they seat relative to the first person, for instance the order A,B,C,D and C,D,A,B are the same(in this situation-circle). In other words the first person has really one choice, so 4! is over counted by four ways, thus the real answer is: (4x3x2x1)/4=3!=6.

Tuugii