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Group Theory

  1. Nov 10, 2013 #1
    1. The problem statement, all variables and given/known data

    In $S_n$, prove that there are ##\frac{n!}{r(n-r)!}## distinct r-cycles.

    2. The attempt at a solution

    I know there are n choose r ways to permute r out of n cycles thus ##\frac{n!}{r!(n-r)!}## but I don't know how they got ##\frac{n!}{r(n-r)!}##?
     
  2. jcsd
  3. Nov 10, 2013 #2

    pasmith

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    Homework Helper

    Two r-cycles are the same if their cycle notations are cyclic permutations of each other. Having chosen our objects, we can avoid such multiple counting by fixing one object to appear the beginning of all the r-cycles. Each permutation of the remaining [itex]r-1[/itex] objects will then generate a distinct r-cycle.
     
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