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| Circular Permutations |
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| When things are arranged in places along a line with first and last place, they form a linear permutation. So far we have dealt only with linear permutations. When things are arranged in places along a closed curve or a circle, in which any place may be regarded as the first or last place, they form a circular permutation. |
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| The permutation in a row or along a line has a beginning and an end, but there is nothing like beginning or end or first and last in a circular permutation. In circular permutations, we consider one of the objects as fixed and the remaining objects are arranged as in linear permutation. |
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| Thus, the number of permutations of 4 objects in a row = 4!, where as the number of circular permutations of 4 objects is (4-1)! = 3!. |
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| The following arrangements of 4 objects O1, O2, O3, O4 in a circle will be considered as one or same arrangement. |
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| Observe carefully that when arranged in a row, O1 O2 O3 O4, O2O3O4 O1, O3O4O1O2, O4O1O2O3 are different permutations. When arranged in a circle, these 4 permutations are considered as one permutation. |
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| The number of circular permutations of n different objects is (n-1)!. |
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Each circular permutation corresponds to n linear permutations depending on where we start. Since there are exactly n! linear
permutations, there are exactly  permutations. Hence, the
number of circular permutations is the same as (n-1)!. |
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| Example 1: |
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| 1. Three boys and three girls are to be seated around a table in a circle. Among them, the boy X does not want any girl as neighbour and girl Y does not want any boy as neighbour. How many such arrangements are possible? |
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| Suggested answer: |
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| The arrangement is as shown in the figure, the boy X will have B2, B3 as neighbours. The girl Y will have G2, G3 as neighbours. The two boys B2, B3 can be arranged in two ways. The two girls G2, G3 can be arranged in two ways. |
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| Hence, the total number of arrangements = 2 x 2 = 4. |
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| Example 2: |
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| In how many different arrangements can 6 gentlemen and 6 ladies sit around a table if |
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| i) there is no restriction and |
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| ii) no two ladies sit side by side? |
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| Suggested answer: |
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| i) Here, the total number = 6 + 6 = 12. |
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| 12 persons can be arranged in circular permutation as (12 - 1)! = 11! ways. |
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| ii) When 6 gentlemen are arranged around a table, there are 6 positions, each being between two gentlemen for 6 ladies, when no two ladies sit side by side. Now, the number of ways in which 6 gentlemen can be seated around a table = (6 - 1)! = 5!. |
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| Then, corresponding to each seating arrangement for the gentlemen, the 6 ladies can be seated in 6! ways. |
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 The required number of arrangements = (5!)(6!) |
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| = (120)(720) = 86400 |
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