Question

In the binomial expansion of $\left(\sqrt[3]{\frac{a}{b}} + \sqrt[3]{\frac{b}{\sqrt{a}}}\right)^{21},$ the term containing same powers of $a$ and $b$ is

• A. $11^{t h}$
• B. $13^{t h}$
• C. $12^{t h}$
• D. $6^{t h}$

Answer 1

Answer: (B)

$t_{r+1}={ }^{21} C_{r}\left(\frac{a}{b}\right)^{\frac{21-r}{3}} \frac{b^{\frac{r}{3}}}{r}$
$={ }^{21} C_{r} a^{\frac{21-r}{3}-\frac{r}{6}} b^{\frac{r}{3}-\frac{21-r}{3}}$
Since, power of $a$ and $b$ are the same $\Rightarrow \frac{21-r}{3}-\frac{r}{6}=\frac{r}{3}-\frac{21-r}{3}$
$\Rightarrow \frac{2(21-r)}{3}=\frac{r}{3}+\frac{r}{6}=\frac{r}{2}$
$84-4 r=3 r \Rightarrow 7 r=84 \Rightarrow r=12$
Hence, the $13^{\text {th }}$ term contains the same powers of $a$ and $b$.