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Math Help - sandwich combinations

  1. #1
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    sandwich combinations

    Choices

    6 bread
    4 meat
    4 cheese
    12 garnishes

    How many different possibilities for sandwiches are there is you choose
    a- 1 bread, 1 meat, 1 cheese, and no garnishes?
    b- 1 bread, 1 meat, 2 DIFFERENT cheeses, and from 0 to 12 garnishes?

    Thanks
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  2. #2
    MHF Contributor matheagle's Avatar
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    multiplication rule
    (a) 6 times 4 times 4
    (b) 6 times 4 times 6 because (4 choose 2 =6 cheeses) the garnishes, well thats huge
    the number of garnishes is the sum of 12 choose x from 0 to 12

    \sum_{x=0}^{12}{12 \choose x}= \sum_{x=0}^{12}{12 \choose x}1^x1^{12-x}= 2^{12}
    via the binomial theorem
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  3. #3
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    Thank you.

    part a=96
    partb=6*4*6*4096=589824?
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  4. #4
    MHF Contributor matheagle's Avatar
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    Yes, if I understood this correctly
    and by the way, who cut the cheese?
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