Theory on Circular Permutations and its application

Circular Permutations

The number of circular permutations of n different things is (n-1)! In this arrangement anti-clockwise and clockwise order of arrangement are considering as distinct.

If anti-clockwise and clockwise order of arrangement are not different, i.e. in case of arrangement of beads in necklace, arrangement of flower in garland etc. then the number of circular permutations of n different things = $$\frac{1}{2} \times \left( {n - 1} \right)!$$

Solved Examples

In how many ways can 6 men be seated around a circular table?
(a)120 (b)24 (c)12 (d)5

Solution: (a)
6 men can be seated around a circular table in (6-1)! = 5! = 5 × 4 × 3 × 2 × 1 = 120 ways.

Find the number of ways in which 10 different beads can be arranged to form a necklace.

Solution:
10 different beads can be arranged in necklace (circular form)
\eqalign{ & = \frac{1}{2} \times \left( {10 - 1} \right)! \cr & = \frac{1}{2} \times \left( 9 \right)!\,\,{\text{ways}} \cr}