Question

Let $\left(1+3 x+2 x^2\right)^6=\displaystyle\sum_{ r =0}^{12} a _{ r } x ^{ r }$. Identify which of the following statement(s) is(are) correct?

• A. The coefficient of $x^{12}$ is $2^{12}$.
• B. The coefficient of $x^{11}$ is $9 \cdot 2^6$.
• C. $\frac{\displaystyle\sum_{ r =0}^6 a _{2 r }}{ a _{12}}=\frac{3^6}{2}$
• D. $\frac{\displaystyle\sum_{ r =1}^6 a _{2 r -1}}{ a _{12}}=2 \cdot 3^6$

Answer 1

Answer: (B), (C)

$\left(1+3 x+2 x^2\right)^6=\displaystyle\sum_{r=0}^{12} a_r x^r$
$\left(1+3 x+2 x^2\right)^6=a_0+a_1 x+a_2 x^2+\ldots \ldots .+a_{12} x^{12}$
Putting $x=1 a_0+a_1+a_2+\ldots \ldots+a_{12}=6^6$
Putting $x=-1 a_0-a_1+a_2+\ldots \ldots .+a_{12}=0$
$a_0+a_2+a_4+\ldots \ldots+a_{12}=3 \cdot 6^5 $
$a_1+a_3+a_5+\ldots \ldots+a_{11}=3 \cdot 6^5$
$\text { coefficients of } x^{12}=2^6 $
$\text { coefficients of } x^{11}:{ }^6 C _6\left(3 x +2 x ^2\right)^6={ }^6 C _6 \cdot x ^6(3+2 x )^6$
$={ }^6 C _6 \cdot{ }^6 C _5 \cdot 3^1 \cdot 2^5 $
$=18 \cdot 2^5=9 \cdot 2^6$