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Math Help - Multinomial expansion

  1. #1
    MHF Contributor alexmahone's Avatar
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    Multinomial expansion

    If we expand the expression

    (x_1+x_2+x_3+x_4)^6

    what will be the largest coefficient that occurs?

    My attempt:

    My hunch is that the coefficient should belong to one of the middle terms.

    \dbinom{6}{2,2,1,1}=\frac{6!}{2!2!1!1!}=180

    Is it right? I'm not sure how to prove it, though.
    Last edited by alexmahone; August 19th 2011 at 04:11 PM.
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  2. #2
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    Re: Multinomial expansion

    Quote Originally Posted by alexmahone View Post
    If we expand the expression

    (x_1+x_2+x_3+x_4)^6

    what will be the largest coefficient that occurs?

    My attempt:

    My hunch is that the coefficient should belong to one of the middle terms.

    \dbinom{6}{2,2,1,1}=\frac{6!}{2!2!1!1!}=180

    Is it right? I'm not sure how to prove it, though.
    Expanding this would be a pain. I would write it as \displaystyle (X_1 + X_2)^6, with \displaystyle X_1 = x_1+ x_2 and \displaystyle X_2 = x_3 + x_4. Then you can use a binomial expansion, and when you back-substitute, you'll end up with more binomials you can expand.
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  3. #3
    MHF Contributor alexmahone's Avatar
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    Re: Multinomial expansion

    Quote Originally Posted by Prove It View Post
    Expanding this would be a pain. I would write it as \displaystyle (X_1 + X_2)^6, with \displaystyle X_1 = x_1+ x_2 and \displaystyle X_2 = x_3 + x_4. Then you can use a binomial expansion, and when you back-substitute, you'll end up with more binomials you can expand.
    The question only asks for the largest coefficient, not for the expansion.
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    Re: Multinomial expansion

    Quote Originally Posted by alexmahone View Post
    The question only asks for the largest coefficient, not for the expansion.
    And yet, expanding is the only way I can think of to get the largest coefficient...
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    Re: Multinomial expansion

    Quote Originally Posted by alexmahone View Post
    If we expand the expression
    (x_1+x_2+x_3+x_4)^6
    what will be the largest coefficient that occurs?
    \dbinom{6}{2,2,1,1}=\frac{6!}{2!2!1!1!}=180
    Is it right?
    Yes, that is correct. Note that is the 'smallest' denominator possible.
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