Circular Permutations

The arrangements we have considered so far are linear. There are also arrangements in closed loops, called circular arrangements.

Consider four persons A, B, C and D, who are to be arranged along a circle. It's one circular arrangement is as shown in adjoining figure.

Shifting A, B, C, D one position in anticlockwise direction we will get arrangements as follows.

Arrangements as shown in figure (I) (II) (III) and (IV) are not different as relative position of none of the four persons A, B, C, D is changed. But in case of linear arrangements the four arrangements are.

Thus, it is clear that corresponding to a single circular arrangement of four different things there will be 4 different linear arrangements. Let the number of different things be n and the number of their circular permutations be x.

Now for one circular permutation, number of linear arrangements is n

For x circular arrangements number of linear arrangements

                                                = nx.                 .............. (1)

But number of linear arrangements of n different things

                                                = n!                  .............. (2)

From (1) and (2) we get

                                           Nx = n! => x = n!/n = (n - 1)!.

Suppose n persons (a₁, a₂, a₃, ......, a_n) are to be seated around a circular table. There are n! ways in which they can be seated in a row. On the other hand, all the linear arrangements

        a₁, a₂, a₃, ........., a_n

        a_n, a₁, a₂, ........., a_n-1

        a_n-1, a_n, a₁, a₂, ........., a_n-2

        ................................................

        ................................................

        a₂, a₃, a₄, ........., a₁