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Math Help - How many permutations are there

  1. #1
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    How many permutations are there

    Let X = (1,2,3,4,5,6,7,8). Determine the number of permutations
    of X which can be expressed as a product of two disjoint cycles, one of length three and the other of length five. Give a brief justification for your solution
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    Quote Originally Posted by usman206 View Post
    Let X = (1,2,3,4,5,6,7,8). Determine the number of permutations
    of X which can be expressed as a product of two disjoint cycles, one of length three and the other of length five. Give a brief justification for your solution
    There are \binom{8}{3} = \frac{8!}{3! 5!} to divide X into two subsets, one of size 3 and one of size 5.

    Given a subset of size 3, how many cycles can we form? There are 3! total orderings of the elements, but each cycle is counted 3 times because there are 3 different places to start listing the cycle. So the total number of cycles is 3! / 3.

    Similarly, we can form 5! / 5 different cycles from a subset of size 5.

    So all together, there are
    \frac{8!}{3! 5!} \cdot \frac{3!}{3} \cdot \frac{5!}{5}
    permutations of the specified type.
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