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Multinomial Theorem

  1. May 17, 2009 #1
    1. The problem statement, all variables and given/known data
    Find the coefficient of the [tex] x^{12}y^{24} [/tex] for [tex] (x^3+2xy^2+y+3)^{18} [/tex].


    2. Relevant equations
    Multinomial theorem, as stated on http://en.wikipedia.org/wiki/Multinomial_theorem


    3. The attempt at a solution
    Using the multinomial theorem in the form of the wikipedia post, I would set [tex] x_1=x^3, x_2=2xy^2, x_3=y, x_4=3 [/tex]. Now I need to find the k's that "match" the coefficients of the term from the problem statement. This will give me a relation between the k's of the following form
    [tex] 3k_1+k_2 = 12, 2k_2+k_3 = 24, and k_1+k_2+k_3+k_4=18 [/tex]. Now I let [tex] k_1 [/tex] vary and record the discrete values [tex] k_2, k_3, k_4 [/tex], but what I'm confused on is why do I sometimes get negative values for [tex] k_4 [/tex]? Do I need to do something different with the relationship of the k's. Thanks in advance.
     
  2. jcsd
  3. May 18, 2009 #2
    You wrote down three restrictions, but there is a fourth as well, namely that all the k_i's must be nonnegative. Looks like k_1 can be 0 or 1, and that's all.
     
  4. May 18, 2009 #3
    cool, so everything else looks good?
     
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