Question

The term independent of $x$ in the expansion of $(1+x{)}^{10}.{(1+\frac{1}{x})}^{10}$ is

• A. ${}^{10}{C}_5$
• B. ${}^{20}{C}_{10}$
• C. ${}^{20}{C}_5$
• D. ${}^{10}{C}_2$

Answer 1

The correct option is B ${}^{20}{C}_{10}$

$T_{r+1}={}^n{C}_r{a}^{n-r}{b}^r$
Applying to the above question $\frac{(x+1){}^{20}}{x^{10}}$, we get
$T_{r+1}=\frac{{}^{20}{C}_r{x}^r}{x^{10}}$

$={}^{20}{C}_r{x}^{r-10}$ ...(i)
For term independent of $x$,
$r-10=0$
$r=10$
Substituting in equation (i), we get
$T_{11}={}^{20}{C}_{10}$