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Math Help - License plate combinations

  1. #1
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    License plate combinations

    I was posed the question
    "how many license plates contain at most 3 numerals followed by exactly 4 distinct letters"

    I figured

    3x26!-22! +
    2x26!-22! +
    1x26!-22!


    How is this expressable or ssimplified and am I even correct
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  2. #2
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    Re: License plate combinations

    Quote Originally Posted by lhurlbert View Post
    I was posed the question
    "how many license plates contain at most 3 numerals followed by exactly 4 distinct letters"

    I figured

    3x26!-22! +
    2x26!-22! +
    1x26!-22!
    That does not work.

    There are _{26}P_4=(26)(25)(24)(23) ways to have exactly 4 distinct letters.

    There are 1+10+100+1000 ways to at most three numerals.
    Don't forget that are ten numerals, and "at most three" means 0, 1, 2, or 3.
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  3. #3
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    Re: License plate combinations

    Hello, lhurlbert!

    I don't see the resoning behind your answer/


    How many license plates contain at most 3 numerals followed by exactly 4 distinct letters?

    It starts with at most 3 digits.

    It can have no digits: 1 way.
    It can have any number from 1 to 999.
    . . Hence, there are 1000 choices for the numerals.

    \text{There are: }\,26\cdot25\cdot24\cdot23 \:=\:358,\!800\text{ possible four-letter "words"}
    . . \text{with distinct letters.}

    \text{Therefore, there are: }\,(1,\!000)(358,\!800) \:=\:358,\!800,\!000\text{ possible license plates.}

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  4. #4
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    Re: License plate combinations

    Quote Originally Posted by lhurlbert View Post
    I was posed the question
    "how many license plates contain at most 3 numerals followed by exactly 4 distinct letters"

    Quote Originally Posted by Soroban View Post
    It starts with at most 3 digits.
    It can have no digits: 1 way.
    It can have any number from 1 to 999.
    . . Hence, there are 1000 choices for the numerals.

    As you can see the answer above differs from the one I gave.
    Here is why: I read the question in such a way that the plates 90ABXT~\&~090ABXT are different.
    In other words: There is 1 way for no digits.
    There are 10 ways for one digit.
    There are 100 ways for two digits.
    There are 1000 ways for three digits.
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