In a wedding banquet, guests are seated in circular table for four. In how many ways can the guests be seated?

We have **learned that the number of permutations **of distinct objects on a straight line is . That is, if we seat the four guests Anna, Barbie, Christian, and Dorcas, on chairs in on a straight line they can be seated in ways (see **complete list**).

However, circular arrangement is slightly different. Take the arrangement of guests *A, B, C, D* as shown in the first figure. The four possible seating arrangements are just a single permutation: in each table, the persons on the left and on the right of each guest are still the same persons. For example, in any of the tables, *B* is on the left hand side of *A* and *D* is on the right hand side of *A*. In effect, the four linear permutations *ABCD*, *BCDA*, *CDAB*, and *DABC* are as one in circular permutation. This means that the number of linear permutations of 4 persons is four times its number of circular permutations. Since the number of all possible permutations of four objects is 4!, the number of circular permutations of four objects is . Continue reading