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Math Help - Probability question involving r-digit numbers

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    Probability question involving r-digit numbers

    What is the probability that a random r-digit number (such that r is greater than or equal to 3) contains at least one 0, at least one 1, and at least one 2.

    An explanation of how to do this would be much appreciated!
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  2. #2
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    Quote Originally Posted by Zennie View Post
    What is the probability that a random r-digit number (such that r is greater than or equal to 3) contains at least one 0, at least one 1, and at least one 2.

    An explanation of how to do this would be much appreciated!
    One way to solve is with inclusion-exclusion.

    I assume leading zeroes are not allowed.

    # of r-digit numbers having no 0's is 9^r, for no 1's it's 8*9^(r-1) and same for no 2's.

    # of r-digit numbers having no 0's and no 1's is 8^r, same for no 0's and no 2's, and for no 1's and no 2's it's 7*8^(r-1).

    The number of r-digit numbers having no 0's, 1's, or 2's is 7^r.

    Apply |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C| \,.

    This gives you the complement set of what you're really trying to count, so subtract it from 9*10^(r-1), alternatively calculate probability and subtract it from 1.
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