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Math Help - Combinatorics Question

  1. #1
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    Combinatorics Question

    I have two questions about probabilities when choosing things from a set.

    Assume you have a bag of x red marbles, y blue marbles, and z green marbles.

    1. If you choose 6 marbles from the bag, without replacement, what is the probability that you get exactly 2 of each type of marble?

    2. If you choose 6 marbles from the bag, with replacement, what is the probability that you get exactly 2 of each type of marble?


    I'm not really sure how to go about solving these.

    Thanks for any help.
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  2. #2
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    You must assume that each of x, y, & z is at least 2.
    { x \choose 2} = \frac {x!}{(x-2)!(2!)} is the number of ways of choosing two reds marbles.
    The total number of ways of choosing six marbles is {x+y+z \choose 6} without replacement.
    So,  \frac {{ x \choose 2}{ y \choose 2}{ z \choose 2}}{{x+y+z \choose 6}}.

    Now you try the next one.
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