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Math Help - probability

  1. #1
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    probability

    the registration numbers of vehicles in certain country use digits 1 to 999. if a visitor has just entered the country, wht is the probbability tht the first car he sees has at least 2 identical digits. answer provided is (29/111). show me the soloution please
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  2. #2
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    Hello, qweiop90!

    The registration numbers of vehicles in certain country use digits 1 to 999.
    If a visitor has just entered the country, what is the probbability that
    the first car he sees has at least 2 identical digits?
    Answer: 29/111
    There are 999 possible numbers.

    The opposite of "at least two identical digits" is "no matching digits."
    . . How many numbers have no matching digits?

    There are: . {\color{blue}9} one-digit numbers.

    For two-digit numbers:
    The first digit is not zero . . . there are 9 choices.
    The second digit can be any digit except the first digit: 9 choices.
    . . There are: . 9\cdot9 = {\color{blue}81} two-digit numbers.

    For three-digit numbers:
    The first digit is not zero . . . there are 9 choices.
    The second digit can be any digit except the first digit: 9 choices.
    The second digit can be any of the remaining 8 digits: 8 choices.
    . . There are: . 9\cdot9\cdot8 = {\color{blue}648} three-digit numgers.

    So there are: . 9 + 81 + 648 \:=\:{\color{blue}738} numbers with no matching digits.

    Hence, there are: . 999 - 738 \:=\:{\color{blue}261} numbers with some matching digits.


    Therefore: . P(\text{some matching digits}) \;=\;\frac{261}{999} \;=\;{\color{red}\frac{29}{111}}

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