Find the coefficient of $x^3$ in the expansion of $(1+x)^3(2+x^2)^{10}$

Question

The coefficient of $x^3$ in the expansion of $(1+x)^3(2+x^2)^{10}$ is

(A) $2^{14}$

(B) $31$

(C) $3\choose{3}$ $+$ $10\choose{1}$

(D) $3\choose{3}$ $+2$$10\choose{1}$

(E)$3\choose{3}$$10\choose{1}$$2^9$

The answer is (A), although I wasn't able to get to any of the answer choices. I tried:

$(1+x)^3=1+3x+3x^2+x^3$

$(2+x^2)^{10}=2^{10}+10\cdot2^9\cdot3x^2+45\cdot 2^8 \cdot 9x^4+...$

So, when the relevant terms are multiplied together, I get $(3x)\cdot (10\cdot2^9\cdot3x^2)+(x^3)\cdot (2^{10})=(90\cdot2^9+2^{10})\cdot x^3=(46\cdot 2^{10})\cdot x^3=(23\cdot2^{11})\cdot x^3$.

Can someone point out my error? Thanks.

Answer 1

The error is here:

$$(2+x^2)^{10}=2^{10}+10\cdot2^9\cdot3x^2+45\cdot 2^8 \cdot 9x^4+\cdots$$

You have an extra factor of $3$ appearing in front of $x^2$. It should read:

$$(2+x^2)^{10}=2^{10}+10\cdot2^9\cdot x^2+45\cdot 2^8 \cdot x^4+\cdots$$

Then, your method gives the correct answer of $3\cdot 10\cdot 2^9 + 2^{10} = 2^{14}$.

Answer 2

You have a $3x^2$ and a $9x^4$ in your $(2+x^2)^{10}$ expansion that should just be $x^2$ and and $x^4$