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Math Help - Counting - Discrete math

  1. #1
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    Counting - Discrete math

    7 woman and 9 man are on the faculty in the math department in the school.

    (1) How many ways are there to select a committee of five members of the department if at least one woman must be on the same committee ?

    (2) how many ways are there to select a committee of five members of the department if at least one woman and at least one man must be on the committee .

    Can some one please help me with this ?
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  2. #2
    Senior Member
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    You are considering a set which has 16 elements, and has a partition in two subsets A and B which have 7 and 9 elements.

    For 1), what you want to know is the number of 5-element subsets of E that have at least one element in A. That means the "first" element x must be in A, you have 7 choices, then every subset with 4 elements in A-\{x\} union \{x\} is a solution.
    So there are 7\times C^{4}_{15} possible committees.

    2) can be done is a similar way.
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