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Math Help - Combinations using Binomial Theorem

  1. #1
    WWU
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    Combinations using Binomial Theorem

    Hello,

    The problem i am having is as follows:

    A deli has 6 choices for bread, 4 choices for cheese, 4 choices for meat, and 12 condiments.

    How many combinations are there if you choose:
    a)one bread, one meat, and one cheese?
    b)one bread, one meat, one cheese, 0-12 condiments?
    c)one bread, 0-2 meats, 0-2 cheeses, and 0-12 condiments?

    I know that part a is 6*4*4 = 96 choices.
    I also know I need to use the binomial theorem to answer pars b and c but I am not exactly sure how to use it correctly.

    Thank you!
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  2. #2
    MHF Contributor matheagle's Avatar
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    IF I understand the 0-12....

    (6)(4)(4)\sum_{x=0}^{12}{12\choose x}

    =(6)(4)(4)\sum_{x=0}^{12}{12\choose x}(1)^{x}(1)^{12-x}

    =(6)(4)(4)\left(1+1\right)^{12}

    =(6)(4)(4)\left(2\right)^{12}
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  3. #3
    MHF Contributor matheagle's Avatar
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    then for (c)

    (6) \sum_{x=0}^2{4\choose x} \sum_{x=0}^2{4\choose x} \biggl(1+1\biggr)^{12}

    =(6) (1+4+6)(1+4+6) (1+1)^{12}
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  4. #4
    WWU
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    Ok, that makes sense. Thank you!
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