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Math Help - Combining Objects in Buckets

  1. #1
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    Question Combining Objects in Buckets

    Howdy!

    I have a set of 8 objects: {A, B, C, D, E, F, G, H}
    I have 6 buckets.
    I must place all 8 objects in any combination of the 6 buckets.

    I can place between 0 and 4 objects in any bucket.
    I must place all objects in a bucket.

    A and B cannot be placed in the same bucket.
    C and D cannot be placed in the same bucket.
    E and F cannot be placed in the same bucket.
    F and G cannot be placed in the same bucket.

    The order of objects in the bucket is irrelevant. (Bucket 1: ACEF is identical to Bucket 1: FECA, i.e.)
    The order of buckets is irrelevant. (Bucket 1: ACEF, Bucket 2: BDEG is identical to Bucket 1: BDEG, Bucket 2: ACEF is identical to Bucket 4: BDEG, Bucket 6: ACEF, i.e.)

    The question is: how many arrangements exist? How do I figure this one out?
    Last edited by banantalis; April 22nd 2012 at 11:34 PM. Reason: lost paragraphing
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  2. #2
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    Re: Combining Objects in Buckets

    Is this one without an easy, mathematical solution?
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  3. #3
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    Re: Combining Objects in Buckets

    Quote Originally Posted by banantalis View Post
    I can place between 0 and 4 objects in any bucket.
    0 and 4 inclusive (meaning 5 choices: 0,1,2,3,4) or "between" 0 and 4 (meaning 3 choices: 1,2,3)?

    AND:
    "F and G cannot be placed in the same bucket."
    Is that correct, or do you mean G and H ?
    Last edited by Wilmer; April 27th 2012 at 07:01 PM.
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  4. #4
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    Re: Combining Objects in Buckets

    Inclusively. {0, 1, 2, 3, and 4}

    And, whups! I meant G and H cannot be taken together.
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  5. #5
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    Re: Combining Objects in Buckets

    Soooo....because of order of buckets and objects being irrelevant,
    there is ONLY 1 way to have 2 buckets at maximum (containing 4) :
    A,C,E,G in any bucket, B,D,F,H in any of remaining 5 buckets....is that your intent?
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  6. #6
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    Re: Combining Objects in Buckets

    Quote Originally Posted by Wilmer View Post
    Soooo....because of order of buckets and objects being irrelevant,
    there is ONLY 1 way to have 2 buckets at maximum (containing 4) :
    A,C,E,G in any bucket, B,D,F,H in any of remaining 5 buckets....is that your intent?
    Yes.
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  7. #7
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    Re: Combining Objects in Buckets

    HOKAY! Une autre question:
    on this 4:3:1 arrangement :
    Code:
       1   2   3   4   5   6
    A  x 
    B          x
    C  x
    D      x
    E  x 
    F      x
    G  x
    H      x
    There are ONLY 4 of these; I show:
    DFH:B ; also possible are:
    BFH:D
    BDH:F
    BDH:H
    But there is NO MORE using these 4 letters.
    Of course, same can be done with A,C,E,G.

    So there are eight 4:3:1 arrangements.

    Oui?
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  8. #8
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    Re: Combining Objects in Buckets

    Quote Originally Posted by Wilmer View Post
    There are ONLY 4 of these; I show:
    DFH:B ; also possible are:
    BFH
    BDH:F
    BDH:H
    But there is NO MORE using these 4 letters.
    Of course, same can be done with A,C,E,G.

    So there are eight 4:3:1 arrangements.

    Oui?
    By my understanding of the problem, that looks right. It would be BDF:H, but yes.
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  9. #9
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    Re: Combining Objects in Buckets

    Quote Originally Posted by banantalis View Post
    By my understanding of the problem, that looks right.
    Well, pretty "discouraging" if YOU are not completely sure what the problem means.

    You answered "yes" to this question earlier:
    Soooo....because of order of buckets and objects being irrelevant,
    there is ONLY 1 way to have 2 buckets at maximum (containing 4) :
    A,C,E,G in any bucket, B,D,F,H in any of remaining 5 buckets....is that your intent?

    However, I goofed: like, A,D,E,G with B,C,F,H also possible...along with another 6 similar cases.

    Here's the 8 cases (I'll assign values; A=1, B=2, .... , H=8):
    1357,2468
    1358,2467
    1367,2458
    1368,2457
    1457,2368
    1458,2367
    1467,2358
    1468,2357

    So, there are 8 ways if 2 buckets are used, not only 1.
    In other words, 8 combinations where 12,34,56,78 are NOT seen in any bucket!

    Next step is to do similar work using 3 buckets...then 4 buckets...keep going.

    But I'm not doing no more...somebody else's turn to have his/her head "spin"
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