Question

The term independent of $x$ is in the expansion of ${(1-x)}^2$.${(x+\frac{1}{x})}^{10}$ is

• A. ${}^{11}{\text{C}}_5$
• B. ${}^{10}{\text{C}}_5$
• C. ${}^{10}{\text{C}}_4$
• D. none of these

Answer 1

The correct option is A ${}^{11}{\text{C}}_5$

$(1-x{)}^2(x+\frac{1}{x}{)}^{10}$

$=(1+x^2-2x)(x+\frac{1}{x}{)}^{10}$

In the above expansion, the term independent of $x$, will come from constant term from second bracket and coefficient of $\frac{1}{x^2}$ in second bracket.
${}^{10}{C}_5+{}^{10}{C}_6$
$={}^{11}{C}_6$
$={}^{11}{C}_{11-6}$
$={}^{11}{C}_5$