# Circular Permutation:

#### What is Circular Permutation?

The different ways to arrange the given "n" distinct objects in a circular manner is called the circular permutation. There are two types of circular permutations.

Note: In circular permutation, one element is always fixed and all other elements are arranged relative to the fixed element.

Type (1): When the clockwise and anticlockwise orders are different then:

The number of circular permutations $$P_n = \frac{nP_r}{r}$$ Where, n = Total number of objects.

r = Number of selected objects.

$$P_n$$ = Circular permutation.

If n = r (selecting all the objects) then,$$P_n = \frac{nP_n}{n}$$ $$= \frac{n \times (n - 1)!}{n}$$ $$P_n = (n - 1)!$$ Here the arrangement (1-2-3 and 1-3-2) is different when we see the images from clockwise and anticlockwise.

Type (2):When the clockwise and anticlockwise orders are the same then:

The number of circular permutations $$P_n = \frac{nP_r}{r}$$ Where, n = Total number of objects.

r = Number of selected objects.

$$P_n$$ = Circular permutation.

If n = r (selecting all the objects) then,$$P_n = \frac{nP_n}{2n}$$ $$= \frac{n \times (n - 1)!}{2n}$$ $$P_n = \frac{(n - 1)!}{2}$$ Here the image (2) is the reflection of image (1), it can not be distinguished. We can say the way of arrangement is the same (1-2-3) whether we see it from clockwise or anticlockwise.

Example(1): Find the number of ways in which three men and three women can sit around a circular table if none of any two women sit together?

Solution: Three men can sit around the circular table in $$(3 - 1)!$$ way = $$2!$$

Three women can sit around the circular table in the space between the men in three chairs if no two women sit together = $$3!$$

Hence required number of ways = $$2! \times 3!$$ = 12 ways.

Example(2): Find the number of ways in which five different colour beads can be arranged to form a necklace?

Solution:Total number of circular permutations$$P_n = \frac{(n - 1)!}{2}$$ $$= \frac{(5 - 1)!}{2}$$ $$= \frac{4!}{2}$$ $$= 12 \ Ways$$