Question

What is circular permutation? Give some examples also.

Answer 1

Hint: A permutation is basically an arrangement of items in a certain order out of which a few of them are taken at a time. In a permutation we count the number of ways the arrangement can occur. Permutations can also be distinguished by looking at the ways in which elements of a set are arranged. Circular permutation is the number of ways to set up $n$ distinct objects or items beside a fixed circle.

Complete step by step answer:
Examples: Find the number of ways in which $8$ stones can be arranged to form a necklace
Sol: First we will fix the position of the $1$ stone. Now, we are left with $7$ stones.These $7$ stones can arrange themselves in ${}^7{P}_7=7!ways$. As we know there is no dependency on the position of stones in an anticlockwise or clockwise manner. The required no. of ways = ${\displaystyle \frac{1}{2}}7!=2,520ways$.

Let us understand circular arrangement by using one more example: In how many ways can $10$ men and $5$ women sit around a circular table such that no $2$ women sit together?
Sol: $10$ men can be seated around a table for $9!$ ways. There are $10$ spaces between the men which can be filled up by the $5$ women in ${}^{10}{P}_5$ ways.
Therefore, total no. of ways of arranging the men and women = $9!\times {}^{10}{P}_5$ ways
$\text{Total no. of ways of arranging the men and women}=9!\times {\displaystyle \frac{10!}{\left(10-5\right)!}}$
$\therefore \text{Total no. of ways of arranging the men and women}=9!\times {\displaystyle \frac{10!}{5!}}$

Note: Remember that there are two cases of circular permutations:
-When clockwise and anticlockwise orders are different, then the total number of circular permutations is given by $\left(n-1\right)!$.
-When clockwise and anticlockwise orders are taken as not different then the total number of circular permutations is given by ${\displaystyle \frac{\left(n-1\right)!}{2!}}$.