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Math Help - Permutation

  1. #1
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    A cafe lets you order a deli sandwich your way. There are 6 choices of bread, 4 choices for meat, 4 choices for cheese, and 12 different garnishes.

    How many different sandwich possibilities are there if you choose:
    One bread, one meat, one cheese, and from 0 to 12 garnishes?
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  2. #2
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    Tell us what have you tried? And where you have trouble?
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  3. #3
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    Ive tried calculating the number of sandwiches from the bread, meat, and cheese, which is 6 x 4 x 4. But im confused of how to calculate the number of sandwiches with the garnishes.
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  4. #4
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    For the garnishes, you have 13 choices, since you can choose
    not to have one.
    It's very unlikely you'd want to mix garnishes together.
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  5. #5
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    Quote Originally Posted by bori02082005 View Post
    Ive tried calculating the number of sandwiches from the bread, meat, and cheese, which is 6 x 4 x 4. But im confused of how to calculate the number of sandwiches with the garnishes.
    You can select the garnishes in 2^{12} ways.
    That is the number of subsets of a set of twelve items.
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  6. #6
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    Thanks Plato,
    a nice alternative to \binom{12}{0}+\binom{12}{1}+\binom{12}{2}+......+\  binom{12}{12}

    1 garnish 'A' allows 2 choices 0 or A.

    2 garnishes 'A' and 'B' allows 2 new additional choices B and (A,B), twice as many choices.

    3 garnishes 'A', 'B' and 'C' allows 4 new additional choices C, (C,A), (C,B), (C,A,B), twice as many choices.

    4 garnishes 'A', 'B', 'C', 'D'.... twice as many choices.

    in general, number of ways of selecting groups of 0 to n =2^n

    I can't begin to imagine a sandwich with 12 garnishes due to a
    sensitive stomach..
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  7. #7
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    Quote Originally Posted by Archie Meade View Post
    I can't begin to imagine a sandwich with 12 garnishes due to a sensitive stomach..
    Reality has absolutely nothing to do with mathematics.
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