Question

What is circular permutation? Give some examples also.

Answer 1

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 = $\dfrac{1}{2}7! = 2,520\,ways$.

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
$\text{Total no. of ways of arranging the men and women} = 9! \times \dfrac{{10!}}{{\left( {10 - 5} \right)!}}$
$\therefore \text{Total no. of ways of arranging the men and women} = 9! \times \dfrac{{10!}}{{5!}}$

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 $\dfrac{{\left( {n - 1} \right)!}}{{2!}}$.