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Question

Passage:
If n is a positive and $$a_{1},\ a_{2},\ a_{3},..a_{m}\in C$$ then
$$(a_{1}+a_{2}+a_{3}+.... +a_{m})^{n}=\displaystyle \Sigma(\frac{n!}{{ n }_{ 1 }!{ n }_{ 2 }!{ n }_{ 3 }!\ldots n_{m}!})a_{1}^{{ n }_{ 1 }}a_{2}^{{ n }_{ 2 }}a_{3}^{{ n }_{ 3 }}....a_{m}^{{ n }_{ m }}$$
where $$n_{1},n_{2},\ n_{3},\ n_{m}$$ are all non negative integers subject to the condition 
$$n_{1}+n_{2}+n_{3}+\ldots+n_{m}=n$$
The coefficient of $$x^{39}$$ in the expansion of $$(1+x+2x^{2})^{20}$$ is


A
5×219
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B
5×230
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C
5×221
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D
5×223
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Solution

The correct option is D $$5\times 2^{21}$$
$$(1+x+2x^2)^{20}$$
$$x^{39}$$ can be written as $$1.x.(2x^2)^{19}$$
Hence coefficient will be
$$\dfrac{20!}{(20-19)1!(20-19)1!(20-1)!}.2^{19}$$
$$=\dfrac{20!}{1!1!19!}.2^{19}$$
$$=20.2^{19}$$
$$=4.5.2^{19}$$
$$=5.2^{21}$$

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