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Math Help - Combinations

  1. #1
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    Combinations

    I hope this is the right place to put this thread.
    How many combinations of three letters taken from the letters A,A,B,B,C,C,D are there?
    I take it as ^7C_3=35 but the answer is 13.
    Thanks for any help.
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  2. #2
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    Hello, arze!

    How many combinations of three letters taken from the letters A,A,B,B,C,C,D are there?
    Your answer would be correct if there were 7 different letters.
    Because of the duplicated letters, I counted the cases rather primitively.


    How many 3-letter combinations have 3 different letters?

    There are 4 different letters available, and we pick 3 of them.
    . . There are: . {4\choose3} \,=\,4 combinations.


    How many 3-letter combinations have 2 of one letter and 1 other letter?

    There 3 choices for the pair (A, B, or C).
    Then there are 3 choices for the single letter.
    . . There are: . 3\cdot3 \,=\,9 combinations.


    Therefore, there are: . 4 + 9 \:=\:13 possible combinations.

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