Question

The sum of coefficients in the binomial expansion of ${(\frac{1}{x}+2x)}^n$is equal to $6561$.The constant term in the expansion is

• A. ${}^8{\text{C}}_4$
• B. $16{\cdot }^8{\text{C}}_4$
• C. ${}^6{\text{C}}_4\cdot 2^4$
• D. none of these

Answer 1

The correct option is B $16{\cdot }^8{\text{C}}_4$

Substituting $x=1$ to get the sum of the binomial coefficients
we get
$(3{)}^n=6561$
Taking $lo{g}_3$ on both the sides, we get
$n=8$
Therefore the above question reduces to
$(x^{-1}+2x{)}^8$
The general term will be
$T_{r+1}={}^8{C}_r{x}^{2r-8}{2}^r$
The term independent of $x$ will be given by
$2r-8=0$
$r=4$
Substituting we get
${}^8{C}_4{2}^4$
$=16\left({}^8{C}_4\right)$