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Math Help - Coefficient and expansions.

  1. #1
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    tehran
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    Coefficient and expansions.

    find the Coefficient of term x2y3 in below expansion:

    expansion : (2+x-y+3z)10

    please write a good description for solution* or how to solve it?
    * for a person who is a dummy in math and combinatorics...
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  2. #2
    Senior Member jakncoke's Avatar
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    Re: Coefficient and expansions.

    80,640. Use the multinomial theorem which says (x_1+...+x_r)^n = \sum_{P(n)}{n \choose k_1,k_2,..,k_r } * (x_1)^{k_1}...(x_n)^{k_r}(Basically sum over all non negative integer partitions of n.)
    the  {n \choose k_1,k_2,..,k_r} = \frac{n!}{k_1!...k_r!}
    basically, it says the powers on the exponents of each term you have in your parenthesis, namely 2,x,y,z must add up to 10.
    so how you can write 10 = 0+0+3+7 which is equivalent to \frac{10!}{0!0!3!7!}2^0*x^0*y^3*z^7
    so since you asked for x^2y^3 this means since we don't see z so the power of z is 0. 2+3+0+x = 10 and so x = 5 (which is the power 2 is raised to)
    so you have \frac{10!}{0!5!3!2!}2^5*x^2*y^3*z^0 = 80640 x^2y^3
    Last edited by jakncoke; February 28th 2013 at 01:12 AM.
    Thanks from mshou
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  3. #3
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    Re: Coefficient and expansions.

    It is - 80640 x^2y^3

    MINOAS
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  4. #4
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    Re: Coefficient and expansions.

    well. thanks and a question:
    isn't there any difference between for instance 3x and x, and -y and y ?
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  5. #5
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    Re: Coefficient and expansions.

    well. thanks and a question:
    isn't there any difference between for instance 3x and x, and -y and y ?
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