Probability (4) circular permutation + combination

Probability (4) circular permutation + combination -...

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Circular permutation
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There are n ! different permutations of n distinct objects. ( arranged in a line ) . How about other kinds of arrangement? Say, if the people are randomly arranged in a circle, how many different ways do we have?
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A B C D E A B D E C A There are 5! different (linear) permutations.
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A B C D E A B D E C A
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B D E C A A B C D E A
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B D E C A A B C D E A B D E C A A B C D E
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A B C D E C A E B D E C D A B D E B C A B D A E C B D E C A A B D E C C A B D E E C A B D D E C A B All are different! All are the same! ( On page 15 of the course note )
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In a circular setting, pattern B-D-E-C-A is “repeated” 5 times in the 5! ways. B D E C A A B D E C C A B D E E C A B D D E C A B All are the same! Similarly, other patterns are also “repeated” 5 times
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Circular permutation Each pattern is “repeated” 5 times. The number of different circular permutations of 5 distinct objects is 5! 5 = 4! The number of different circular permutations of n distinct objects is (n-1)!
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Examples ( On page 15 of the course note ) Question 5: If N people, including A and B, are randomly arranged in a line , what is the number of permutations that A and B are next to each other ? What would the number of circular permutations be if the people are randomly arranged in a circle ? Ans: (N-1)!2! Ans: [(N-1)-1]!2!
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with some indistinguishable objects
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Probability (4) circular permutation + combination -...

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