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Math Help - Coefficient of expansion

  1. #1
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    Coefficient of expansion

    How do I find the coefficient of x^{11} in the expansion of (2 + x^2 - 4x^3)^{15}?
    Last edited by CaptainBlack; March 15th 2009 at 01:36 AM. Reason: fixed LaTeX
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  2. #2
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    Hello, chris414!

    How do I find the coefficient of x^{11} in the expansion of (2 + x^2 - 4x^3)^{15} ?

    Each term in the expansion has the form: . {15\choose p,q,r}(2)^p\,\left(x^2\right)^q\,\left(\text{-}4x^3\right)^r
    . . where p+q+r \:=\:15

    We want the terms in which the sum of the exponents on x's total 11.
    . . That is: . 2q + 3r \,=\,11
    This happens for two cases: . (q,r) \:=\:(1,3),\:(4,1)


    The two terms are:

    . . {15\choose11,1,3}2^{11}(x^2)^1(\text{-}4x^3)^3 \:=\:5460\cdot2048\cdot x^2\cdot(\text{-}64x^9)\:=\:-715,\!653,\!120\,x^{11}<br />

    . . {15\choose10,4,1}2^{10}(x^2)^4(\text{-}4x^3)^1 \:=\:15,\!015\cdot 1024\cdot x^8\cdot(\text{-}4x^3) \:=\:-61,\!501,\!440\,x^{11}


    Therefore, the term is: . \boxed{-777,\!154,\!560}\,x^{11}

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