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Math Help - Counting using inclusion/exclusion principle.

  1. #1
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    Counting using inclusion/exclusion principle.

    I am interested in arrangements of MATHEMATICS
    with

    both T's before both A's or
    both A's before both M's or
    both M's before the E


    I define the following sets

    A_1 = {arrangements with both T's before both A's}

    A_2 = {arrangements with both A's before both M's}

    A_3 = {arrangements with both M's before E}

    I am able to find the cardinality of everything else but A_1 \cap A_3

    i.e. arrangements with both T's before both A's and both M's before E.

    I know I want to pick four of the 11 spots (11 choose 4)

    and arrange H,I,C,S in those spots (4! ways)

    But I dont know whaat to do next, I can't place only the T's in the two left most spots because certainly go T,M,M,E,T.

    Thank you for your help.
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  2. #2
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    Re: Counting using inclusion/exclusion principle.

    Quote Originally Posted by Jame View Post
    I am interested in arrangements of MATHEMATICS with
    both T's before both A's or
    both A's before both M's or
    both M's before the E
    I define the following sets
    A_1 = {arrangements with both T's before both A's}
    A_2 = {arrangements with both A's before both M's}
    A_3 = {arrangements with both M's before E}
    I am able to find the cardinality of everything else but A_1 \cap A_3
    Surely \|A_1 \cap A_3\|=\binom{11}{4}(4!)\binom{7}{4}.

    i.e. Place the \{H,I,C,S\} and arrange; select the four places to place \{T,T,A,A\}(only one way); finally put the \{M,M,E\} is the remaining three places.
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