Explanation

(b)
We have, $T_{r + 1} = \ ^{21}C_{r}\left( \frac{a^{\frac{1}{3}}}{b^{\frac{1}{6}}} \right)^{21 - r}\left( \frac{b^{\frac{1}{2}}}{a^{\frac{1}{6}}} \right)^{r}$
$= \ ^{21}C_{r}\frac{a^{7 - (\frac{r}{3})}}{b^{\frac{7}{2 - \frac{r}{6}}}}.\frac{b^{\frac{r}{2}}}{a^{\frac{r}{6}}}$
$= \ ^{21}C_{r}a^{7 - (\frac{r}{2})}b^{\frac{2r}{3 - \frac{7}{2}}}$
Since, exponents of $a$ and $b$ in the $(r + 1)$th term are equal
$\therefore\ 7 - \frac{r}{2} = \frac{2r}{3} - \frac{7}{2}$
$\Rightarrow \ \frac{21}{2} = \frac{7}{6}r\ \Rightarrow r = 9$