How do I find the coefficient of $\displaystyle x^{11}$ in the expansion of $\displaystyle (2 + x^2 - 4x^3)^{15}?$
Hello, chris414!
How do I find the coefficient of $\displaystyle x^{11}$ in the expansion of $\displaystyle (2 + x^2 - 4x^3)^{15}$ ?
Each term in the expansion has the form: .$\displaystyle {15\choose p,q,r}(2)^p\,\left(x^2\right)^q\,\left(\text{-}4x^3\right)^r$
. . where $\displaystyle p+q+r \:=\:15$
We want the terms in which the sum of the exponents on $\displaystyle x$'s total 11.
. . That is: .$\displaystyle 2q + 3r \,=\,11$
This happens for two cases: .$\displaystyle (q,r) \:=\:(1,3),\:(4,1)$
The two terms are:
. . $\displaystyle {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}
$
. . $\displaystyle {15\choose10,4,1}2^{10}(x^2)^4(\text{-}4x^3)^1 \:=\:15,\!015\cdot 1024\cdot x^8\cdot(\text{-}4x^3) \:=\:-61,\!501,\!440\,x^{11}$
Therefore, the term is: .$\displaystyle \boxed{-777,\!154,\!560}\,x^{11} $