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Math Help - Combination

  1. #1
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    Combination

    Hello,
    I have the answers to the following two questions, but I'm stumped as to why the answers are calculated this way:

    Q.1) There are six comics: A, B, C, D, E, F; How many ways are there to select six comics?
    Answer: C(6+6-1, 6-1)

    Q.2) There are 20 balls. 6 red, 6 green, 8 purple. In how many ways can we select five balls if balls of the same color are considered identical?
    Answer: C(3+5-1, 5)

    What I don't understand is why we subtract 1 from 6 in the first question while leaving 5 just the way it is in the 2nd question.

    Any help would be appreciated. Thanks!
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  2. #2
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    Re: Combination

    Quote Originally Posted by pauliwago View Post
    Hello,
    I have the answers to the following two questions, but I'm stumped as to why the answers are calculated this way:
    Q.1) There are six comics: A, B, C, D, E, F; How many ways are there to select six comics?
    Answer: C(6+6-1, 6-1)
    Q.2) There are 20 balls. 6 red, 6 green, 8 purple. In how many ways can we select five balls if balls of the same color are considered identical?
    Answer: C(3+5-1, 5)

    What I don't understand is why we subtract 1 from 6 in the first question while leaving 5 just the way it is in the 2nd question.
    Any help would be appreciated. Thanks!
    These are also known as multi-selections.
    \binom{N+k-1}{N} ways to select N items from k different types.

    There twenty kinds s of doughnuts, how many ways can we select a dozen :
    \binom{12+20-1}{12}.
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