Question

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

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

Answer 1

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

$(1-2x+x^2)(x+\frac{1}{x}{)}^{10}$
$=(x+\frac{1}{x}{)}^{10}-2x(x+\frac{1}{x}{)}^{10}+x^2(x+\frac{1}{x}{)}^{10}$
For $(x+\frac{1}{x}{)}^{10}$
$T_{r+1}={}^{10}{C}_r{x}^{10-2r}$
Then the coefficient of term independent of x will be the sum of the coefficient of x independent terms in $(x+\frac{1}{x}{)}^{10}$, $2x(x+\frac{1}{x}{)}^{10}$ and $x^2(x+\frac{1}{x}{)}^{10}$
Therefore
$(x+\frac{1}{x}{)}^{10}-2x(x+\frac{1}{x}{)}^{10}+x^2(x+\frac{1}{x}{)}^{10}$
$={}^{10}{C}_5+0+{}^{10}{C}_6$ ...(0 since there are no independent terms in $2x(x+\frac{1}{x}{)}^{10}$)
$={}^{10}{C}_5+{}^{10}{C}_6$
$={}^{11}{C}_5$