Question

The sum of the coefficient in the expansion of ${\left(\frac{1}{x}+2x\right)}^n$ is $6561$. Find the coefficient independent of $x$

• A. $16{}^8{C}_4$
• B. ${}^8{C}_4$
• C. ${}^8{C}_5$
• D. None of these

Answer 1

The correct option is A

$16{}^8{C}_4$

Explanation for correct option:

Given that the sum of the coefficient in the expansion of ${\left(\frac{1}{x}+2x\right)}^n$ is $6561$

Let $x=1$

$$\begin{gathered} \Rightarrow {\left(1+2\right)}^n=6561 \\ \Rightarrow 3^n=3^8 \\ \Rightarrow n=8 \end{gathered}$$

The general term is $T_{r+1}={{}^{8}C}_r{\left(\frac{1}{x}\right)}^{8-r}{\left(2x\right)}^r={{}^{8}C}_r{\left(x\right)}^{r-\left(8-r\right)}{\left(2\right)}^r={{}^{8}C}_r{\left(x\right)}^{2r-8}{\left(2\right)}^r$

The term is independent of $x$

$$\begin{gathered} \therefore 2r-8=0 \\ \Rightarrow r=4 \end{gathered}$$

Therefore the coefficient is

$$\begin{gathered} ={{}^{8}C}_4{\left(\frac{1}{x}\right)}^{8-4}{\left(2x\right)}^4 \\ =16{{}^{8}C}_4 \end{gathered}$$

Hence option A is correct.