Question

In the expansion of ${(1+3x+2x^2)}^6$is , the coefficient of $x^{11}$ is

• A. $144$
• B. $288$
• C. $216$
• D. $576$
• E. $3.2^{11}$

Answer 1

The correct option is D

$576$

Explanation For The Correct Option:

Finding the coefficient of $x^{11}$:

Rewriting the equation as,

${\left[\left(1+3x\right)+\left(2x^2\right)\right]}^6$

Expanding the expression we get

$\Rightarrow {}^6{C}_r{\left(1+3x\right)}^{6-r}{\left(2x^2\right)}^r....\left(i\right)\left[\because {\left(a+b\right)}^n{=}^n{C}_r{\left(a\right)}^{n-r}{\left(b\right)}^r\right]$

For finding the value of $r$, let's consider $(1+3x)$as $x$

Simplifying the power of $x$ in equation $\left(i\right)$ we get

$x^{(6-r)+\left(2r\right)}=x^{6+r}$

If $x^{6+r}=x^{11}\Rightarrow r=5$

For finding the coefficient of $x^{11}$ Substitute $x=1$ and $r=5$ into $\left(i\right)$

$$\begin{gathered} \Rightarrow {}^6{C}_5\times 3\times 2^5 \\ \Rightarrow \frac{6!}{5!(6-5)!}\times 3\times 2^5 \\ \Rightarrow 576 \end{gathered}$$

Hence, option (D) is the correct answer.