Question

Passage:
If n is a positive and $a_1,a_2,a_3,..a_m\in C$ then
$(a_1+a_2+a_3+....+a_m{)}^n=\Sigma \left(\frac{n!}{n_1!n_2!n_3!\dots n_m!}\right)a_1^{n_1}{a}_2^{n_2}{a}_3^{n_3}....a_m^{n_m}$
where $n_1,n_2,n_3,n_m$ are all non negative integers subject to the condition
$n_1+n_2+n_3+\dots +n_m=n$
The coefficient of $x^{39}$ in the expansion of $(1+x+2x^2{)}^{20}$ is

• A. $5\times 2^{19}$
• B. $5\times 2^{30}$
• C. $5\times 2^{21}$
• D. $5\times 2^{23}$

Answer 1

Answer: (C)

$5\times 2^{21}$

$(1+x+2x^2{)}^{20}$
$x^{39}$ can be written as $1.x.(2x^2{)}^{19}$
Hence coefficient will be
$\frac{20!}{(20-19)1!(20-19)1!(20-1)!}{.2}^{19}$
$=\frac{20!}{1!1!19!}{.2}^{19}$
$={20.2}^{19}$
$={\mathrm{4.5.2}}^{19}$
$={5.2}^{21}$