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Math Help - Combinations removing duplications and repetition

  1. #1
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    Combinations removing duplications and repetition

    Is there a general formula which can tackle the following:
    given n objects and r spots, I want to calculate the number of possible combinations accounting for duplicates and repetition.

    eg. n = 4, r = 3 (AAAB)
    possible combinations:
    AAA
    AAB
    result is 2

    eg. n = 4, r = 3 (AABC)
    possible combinations:
    AAB
    AAC
    ABC
    result is 3

    eg. n = 5, r = 3 (AABBC)
    possible combinations:
    AAB
    AAC
    ABB
    ABC
    BBC
    result is 5

    Thank you
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  2. #2
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    Re: Combinations removing duplications and repetition

    Quote Originally Posted by s4276560 View Post
    Is there a general formula which can tackle the following:
    given n objects and r spots, I want to calculate the number of possible combinations accounting for duplicates and repetition.

    eg. n = 4, r = 3 (AAAB)
    possible combinations:
    AAA
    AAB
    result is 2

    eg. n = 4, r = 3 (AABC)
    possible combinations:
    AAB
    AAC
    ABC
    result is 3

    eg. n = 5, r = 3 (AABBC)
    possible combinations:
    AAB
    AAC
    ABB
    ABC
    BBC
    result is 5
    I think you are expecting a simple formula for this but there is none.
    For one, we need to know not only how different kinds we have but also how many of each kind.
    Here is an example.
    \{A,A,A,B,B,C\},~n=6<~\&~r=3.
    Expand (1+x)(1+x+x^2)(1+x+x^2+x^3)=1+3x+5x^2+6x^3+5x^4+3x  ^5+x^6.
    Looking at the coefficients we can answer your question.
    6x^3 tells us there are six possible combinations of three.
    5x^4 tells us there are five possible combinations of four, etc.
    Whole courses have been given on counting problems.
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