Permutations and Combinations

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6. Circular Permutations

By fundamental principle of multiplication, we have seen that \(n\) person can sit on \(n\) chairs, which are in a line, in  \(n!\) ways. Now suppose there are 4 chairs around a circle and are equally spaced and there are 4 persons to sit on these chairs. Now the first person does not have 4 ways to select his chair, has only one way to select his chair as all the positions around a circle are identical. All the four positions in the diagram are identical.


When the first person occupies any position, remaining three positions become distinct and remaining three persons can occupy these positions in 3! ways.

In other words, we know that a circular arrangement remains unchanged if every object is shifted uniformly one place to the right or left. The following two arrangements are identical.


But if we fix position of one object, then remaining three positions become distinct and can be filled in 3! ways. Hence number of circular permutations of  \(n\)  distinct objects =  \((n - 1)!\)

Suppose 5 persons A, B, C, D and E sit around a table having 5 equally spaced chairs. Then the total number of seating arrangements = 4! = 24. In these arrangements clockwise and anticlockwise permutations are taken distinct. If we consider a different case of making a necklace by using 5 different flowers A, B, C, D and E. Then a clockwise arrangement (say ABCDE) and a anticlockwise arrangement (say EDCBA) will be identical.

Hence number of circular permutations of  \(n\)  distinct objects = \(\cfrac{{(n - 1)!}}{2}\), if there is no difference between clock wise and anticlockwise permutations.

Number of circular permutations taking \(r\) distinct objects out of \(n\)distinct objects = \(^n{{\rm{C}}_r} \times (r - 1)!\)

Example  01: 7 persons out of a group of 10 are selected and made to sit around a round table having 7 chairs. Find the total number of seating arrangement.

7 persons out of 10 can be selected in \(^{10}{{\rm{C}}_7}\) ways.
Number of seating arrangement = \(^{10}{{\rm{C}}_7} \times 6!\)= 86400 ways

Example 02: In how many ways 5 men and 5 women can sit at a round table if:
(a) No two women sit together
(b) Not more than 4 women sit together

(a) 5 men can sit first in 4! ways. Now the remaining 5 distinct seats will be occupied by 5 women in 5! ways. Required number of ways = 4!×5!
(b) In this case 5 women cannot sit together. Hence required number of ways 
total number of ways – number of seating arrangement when all 5 women are sitting together= 9! – (5!)×(5!)

Example 03: How many ways different garlands can be formed using 10 different flowers?

In case of flowers, clock wise and anti-clock wise permutations are treated as same. Hence number of different garlands = \(\frac{{(10 - 1)!}}{2} = \frac{{9!}}{2}\)

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