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Multinomial coefficient question.

  1. Apr 9, 2007 #1
    let r>1 which term in (x1+...+xk)^rk has the greatest coefficient?
    well i have this equation:
    [tex](x_1+x_2+...+x_k)^{rk}=\sum_{n_1+n_2+...+n_k=rk}\left(\begin{array}{cc}rk\\\ n_1,n_2,...,n_k\end{array}\right)x^{n_1}...x^{n_k}[/tex]
    well if we notice that (n_1+...+n_k)/k=r then the maximum coefficient is achieved when n_1=n_2=...=n_k=r, but the only way i can see how show that this is true is with lagrange multipliers, and i havent yet used this method in my calclulus classes so i guess there's a combinatorial solution here. any one care to hint me this method?

    thanks in advance.
     
    Last edited: Apr 9, 2007
  2. jcsd
  3. Apr 9, 2007 #2
    well i think i solved it.
    if one of n_k's is smaller than r then there must be another one that is bigger than r and so we will have the coeffiecient smaller than the one achieved by n1=...=nk=r.
    this is why we get that this must be hthe maximum coefficient.
     
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