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  1. #1
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    choose x distinguishable devices from a set

    another question concerning the binomial coefficient from my "discrete mathematics and its applications" book:

    "there are 8 sorts of distinguishable devices. How many ways are there to choose a dozen devices with at least 3 of sort A and no more than 2 of sort B." solution: 9724

     {7-1+7 \choose 7} + {2-1+8 \choose 2} = 1716+36

    a dozen = 12. 12 - 3 ( at least 3 of sort A ) = 9. 9-2 = 7 - so i can choose 7 out of 7 sorts, this is the first part of the sum. the second part of the sum is my second choise of 2 out of 8 distinguishable devices. i sum them up because there is no intersection.

    what did i do wrong ?
    Last edited by dayscott; December 13th 2009 at 10:45 AM.
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  2. #2
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    Once again, the fastest way to this answer is by way of generating functions.
    Find the coefficient of x^{12} the expansion of \left( {1 + x + x^2 } \right)\left( {\sum\limits_{k = 3}^{12} {x^k } } \right)\left( {\sum\limits_{k = 0}^9 {x^k } } \right)^6 .

    But you seem not to be using generating functions in the class.


    So here is tedious way.
    \binom{9+8-1}{9} is the number of ways to select a dozen including at least three of type A.

    \binom{6+8-1}{6} is the number of ways to select a dozen including at least three of type A and least three of type B.


    Thus, \binom{9+8-1}{6}-\binom{6+8-1}{6}=9724
    is the number of ways to select a dozen including at least three of type A but no more than two of type B.
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  3. #3
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    Quote Originally Posted by Plato View Post
    Once again, the fastest way to this answer is by way of generating functions.
    Find the coefficient of x^{12} the expansion of \left( {1 + x + x^2 } \right)\left( {\sum\limits_{k = 3}^{12} {x^k } } \right)\left( {\sum\limits_{k = 0}^9 {x^k } } \right)^6 .

    But you seem not to be using generating functions in the class.

    .

    in class we had tasks like: (x+y)^{25} whats the coefficient of x^{13} ?

    but .. i am not able to extrapolate in my brain how to use this in this excercise, if it is not to hard, may be you can try to explain?
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