Coefficient of expansion

**chris414** · March 14th 2009, 11:26 PM

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

**Soroban** · March 15th 2009, 06:30 AM

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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