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Math Help - Permutations of a Mulit-Set

  1. #1
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    Permutations of a Mulit-Set

    Determine the number of permutations of the multiset T = {3*a, 4*b, 2*c} where, for each type of letter, the letters of the same type do not all appear consecutively. (Thus aabbacbcb is allowed, but accbababb is not.)

    I know the inclusion-exclusion principle is ideal for this situation, but I don't know how to do this..

    Thanks in advance!!
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  2. #2
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    Quote Originally Posted by jzellt View Post
    Determine the number of permutations of the multiset T = {3*a, 4*b, 2*c} where, for each type of letter, the letters of the same type do not all appear consecutively. (Thus aabbacbcb is allowed, but accbababb is not.)
    Let \mathcal{A} be the strings in which all a's are together.
    That number is \#\mathcal{A}
    Now count, \#\mathcal{A}+\#\mathcal{B}+\#\mathcal{C}-\#\mathcal{A\&B}-\#\mathcal{A\&C}-\#\mathcal{B\& C}+\#\mathcal{A\& B\& C}

    You will have counted the cases in which at least one group of letters are together. Subtract.

    I will get you started with \#\mathcal{A}=\dfrac{7!}{4!\cdot 2!} and \#\mathcal{A\& B}=\dfrac{4!}{ 2!}
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