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Math Help - Counting (Permutations/Combinations)

  1. #1
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    Unhappy Counting (Permutations/Combinations)

    A group of 10 is going to be selected from a pool of 8 men and 8 women...
    In how many ways ways can the selection be carried out if:
    a) we choose 10 people at random?
    b) there must be 5 men and 5 women?
    c) there must be more women than men?


    Also, if you have time, can you answer:
    - How many strings of 6 decimaal digit.s have exactly three digits that are 4's?

    I suck at this problems and help would be great! Thank you!
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by achua7 View Post
    A group of 10 is going to be selected from a pool of 8 men and 8 women...
    In how many ways ways can the selection be carried out if:
    a) we choose 10 people at random?
    b) there must be 5 men and 5 women?
    c) there must be more women than men?


    Also, if you have time, can you answer:
    - How many strings of 6 decimaal digit.s have exactly three digits that are 4's?

    I suck at this problems and help would be great! Thank you!
    You only need to know that there are \binom{n}{k} ways to choose a k-element subset from an n-element set.

    Thus you get
    a) \binom{10+10}{10}=\binom{20}{10}=184'756

    b) \binom{10}{5}\cdot\binom{10}{5}=\binom{10}{5}^2=63  '504

    c) Let m be the number of men chosen, the corresponding number of women must then be 10-m; to satisfy the condition that m<10-m, m can only assume the values m=0, 1, 2, 3, or 4. Hence, by summing all these cases we get

    \sum_{m=0}^4\binom{10}{m}\cdot\binom{10}{10-m}=\sum_{m=0}^4\binom{10}{m}^2=60'626

    As to the number of strings of digits (not numbers) of length 6 that contain exactly 3 times the digit 4: you can choose the position of the 3 digits that are 4 in \binom{6}{3} ways, and each of these can be combined with one of the 9^2 ways to choose the remaining 2 digits, different from 4.
    Thus you get \binom{6}{3}\cdot 9^2=1'215 possibilities.
    Note that there is a difference depending on whether we are talking here of the number of strings of length 6 or of numbers with 6 digits, because in the latter case, the first digit would not be allowed to be 0.
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