Question

The coefficient of the term independent of $x$ in the expansion of ${\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)}^{10}$ is equal to

• A. $10$
• B. $252$
• C. $20$
• D. $256$

Answer 1

Answer: (B)

$\therefore$ The general term in the expansion of
$\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)$ is $T_{r+1}{=}^{10}{C}_r{(\sqrt{x})}^{10-r}{\left(\frac{1}{\sqrt{x}}\right)}^r$
${=}^{10}{C}_r{x}^{\frac{10-r}{2}}\left(x^{-\frac{r}{2}}\right)$
${=}^{10}{C}_r{x}^{\frac{10-2r}{2}}$
For independent of x, Put
$\frac{10-2r}{2}=0,$ we get $r=5$ s
$\therefore$ Required coefficient ${=}^{10}{C}_5$
$=\frac{10\times 9\times 8\times 7\times 6}{5\times 4\times 3\times 2\times 1}$ $=252$