Question

The term NOT containing $x$ in the expansion of ${\left(1-x\right)}^2{\left(x+\frac{1}{x}\right)}^{10}$ is

• A. ${}^{11}{C}_5$
• B. ${}^{10}{C}_5$
• C. ${}^{10}{C}_4$
• D. ${}^{10}{C}_7$

Answer 1

Answer: (A)

${\left(1-x\right)}^2{\left(x+\frac{1}{x}\right)}^{10}$
Term independent of x is
${}^2{C}_0(-x{)}^2\times {}^{10}{C}_4(x{)}^4{\left(\frac{1}{x}\right)}^6+{}^2{C}_2{\left(-x\right)}^0\times {}^{10}{C}_{5}x{}^5{\left(\frac{1}{x}\right)}^5$
$=1\times {}^{10}{C}_4+1{\times }^{10}{C}_5={}^{10}{C}_4+{}^{10}{C}_5={}^{11}{C}_5$