What is the Formula for Circular Permutation ?

Here, we will learn circular permutation and the formula for circular permutation with examples.

Let’s begin –

Circular Permutation :

If there are 4 different things, then for each circular arrangement number of linear arrangement is 4.

Similarly, if n different things are arranged along a circle, for each circular arrangements number of linear arrangement is n.

Therefore, the number of linear arrangements of n different things is n \(\times\) (number of circular arrangements of n different things). Hence the number of circular arrangements of n different things is –

\(1\over n\) \(\times\) (number of linear arrangements of n different things) = \(n!\over n\) = \((n – 1)!\)

Formula :

therefore note that, the formula for circular permutation is –

(i) The number of circular permutation of n different things taken all at a time is:

(n – 1)!

If the clockwise and anticlockwise circular permutations are considered to be same, then it is:

\((n – 1)!\over 2\)

(ii) Number of circular permutations of n different things taking r at a time distinguishing clockwise and anticlockwise arrangements is :

\(^{n}P_r\over r\)

Example : A person invites a group of friends at dinner. They sit

(i) 5 on one round table and 5 on other round table

(ii) 4 on one round table and 6 on other round table

Find the number of ways in each case in which he can arrange the friends.

Solution : (i) The number of ways of selection of 5 friends for the first table is \(^{10}C_5\). Remaining 5 friends are left for the second table.

The total number of permutation of 5 friends at a round table is 4!. Hence, the total number of arrangements is \(^{10}C_5\) \(\times\) 4! \(\times\) 4!

(ii) The number of ways of selection of 6 friends is \(^{10}C_6\). Remaining 4 friends are left for the second table.

The number of ways of permutation of 6 friends at a round table is 5!. The number of ways of 4 friends at a round table is 3!

Hence, the total number of arrangements is \(^{10}C_6\) \(\times\) 5! \(\times\) 3!

Hope you learnt the formula for circular permutation, learn more concepts of permutation and combination and practice more questions to get ahead in the competition. Good luck!

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