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Math Help - Multinomial Coefficient

  1. #1
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    Multinomial Coefficient

    Use the Multinomial Theorem to find the coefficient of x^7*y^8 in
    (2*x^3 + 3*x^2*y + 5*x*y^2 + 7*y^3)^5.
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  2. #2
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    Quote Originally Posted by noles2188 View Post

    Use the Multinomial Theorem to find the coefficient of x^7y^8 in (2x^3 + 3x^2y + 5xy^2 + 7y^3)^5.
    the coefficient is \sum \binom{5}{i,j,k,\ell}2^i3^j5^k7^{\ell}, where the sum is taken over all integer solutions of the system: \begin{cases} 3i +2j+k=7 \\ j+2k+3\ell = 8 \end{cases}, and 0 \leq i,j,k,\ell \leq 5. note

    that the condition i+j+k+\ell=5 is a result of the above equations. also see that we actually have: 0 \leq i \leq 2, \ 0 \leq j \leq 3, \ 0 \leq k \leq 4, \ 0 \leq \ell \leq 2. one

    way to find all solutions is to consider four cases j=0, 1, 2, 3. it will take less than 5 minutes of your time!
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  3. #3
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    I'm confused. Where do i plug in the j=0,1,2,3?
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