Question

If the sum of the binomial coefficients of the expansion ${(2x+\frac{1}{x})}^n$ is equal to 256, then the term independent of x is

• A. 1120
• B. 1020
• C. 512
• D. None of these

Answer 1

The correct option is A

1120

1120 Suppose (r+1)th term in the given expansion is independent of x.

Then, we have

$T_{r+1}={}^n{C}_r(2x{)}^{n-r}{\left(\frac{1}{x}\right)}^r$

$={}^n{C}_r{2}^{n-r}{x}^{n-2r}$

For this term to be independent of x, we must have

n-2r =0

$\Rightarrow r=\left(\frac{n}{2}\right)$

$\therefore$ Required term $={}^n{C}_{n/2}{2}^{n-n/2}=\frac{n!}{\left[\right(n/2)!{]}^2}{.2}^{n/2}$

We know:

Sum of the given expansion =256

Thus, we have

$2^n.1^n=256\Rightarrow n=8$

$\therefore$ Required term $=\frac{8!}{\left(4\right)!\left(4\right)!}{.2}^4=1120$