Question

The coefficient of the term independent of $x$ in the expansion of ${(\sqrt{\frac{x}{3}}+\frac{\sqrt{3}}{x^2})}^{10}$ is

• A. $\frac{5}{9}$
• B. $\frac{5}{3}$
• C. $\frac{1}{3}$
• D. $\frac{4}{3}$

Answer 1

The correct option is B $\frac{5}{3}$

$T_{r+1}={}^n{C}_r{a}^{n-r}{b}^r$
Applying to the above question, we get
$T_{r+1}={}^{10}{C}_r(\frac{x}{3}{)}^{(5-\frac{r}{2})}{x}^{-2r}{3}^{\frac{r}{2}}$
$={}^{10}{C}_r{x}^{(5-\frac{5r}{2})}{3}^{r-5}$ ...(i)
For term independent of $x$
$\Rightarrow 5-\frac{5r}{2}=0$
$\Rightarrow r=2$
Substituting in (i). we get
$T_3={}^{10}{C}_2{3}^{-3}$ $=\frac{9.10}{2!.27}$ $=\frac{5}{3}$