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!