Question

In the expansion of ${\left(\frac{x+1}{x^{\frac{2}{3}}-x^{\frac{1}{3}}+1}\frac{x-1}{x-x^{\frac{1}{2}}}\right)}^{10}$, the terms which does not contain $x$, is equal to

• A. ${}^{10}{C}_0$
• B. ${}^{10}{C}_7$
• C. ${}^{10}{C}_4$
• D. ${}^{10}{C}_5$

Answer 1

The correct option is C ${}^{10}{C}_4$

Given,

${(\frac{x+1}{x^{\frac{2}{3}}-x^{\frac{1}{3}}+1}-\frac{x-1}{x-x^{\frac{1}{2}}})}^{10}$

$={\left(\right(x^{\frac{1}{3}}+1)-\frac{(\sqrt{x}-1)(\sqrt{x}+1)}{\sqrt{x}(\sqrt{x}-1)})}^{10}$

$={(x^{\frac{1}{3}}+1-1-x^{\frac{-1}{2}})}^{10}$

$=(x^{\frac{1}{3}}-x^{\frac{-1}{2}})$

The general term is

$(\frac{10}{r})x^{\frac{r}{3}}{x}^{-\frac{(10-r)}{2}}$

$\therefore \frac{r}{3}-\frac{(10-r)}{2}=0$

we get, $r=6$

$(\frac{10}{6})=(\frac{10}{4})=210$