Which term in the expansion of
Question:

Which term in the expansion of $\left\{\left(\frac{x}{\sqrt{y}}\right)^{1 / 3}+\left(\frac{y}{x^{1 / 3}}\right)^{1 / 2}\right\}^{21}$ contains $x$ and $y$ to one and the same power?

Solution:

Suppose $T_{r+1}$ th term in the given expression contains $x$ and $y$ to one and the same power.

Then,

$T_{r+1}$ th term is

${ }^{21} C_{r}\left[\left(\frac{x}{\sqrt{y}}\right)^{1 / 3}\right]^{21-r}\left[\left(\frac{y}{x^{1 / 3}}\right)^{1 / 2}\right]^{r}$

$={ }^{21} C_{r}\left(\frac{x^{(21-r) / 3}}{x^{r / 6}}\right)\left(\frac{y^{r / 2}}{y^{(21-r) / 6}}\right)$

$={ }^{21} C_{r}(x)^{7-r / 2}(y)^{2 r / 3-7 / 2}$

Now, if $x$ and $y$ have the same power, then

$7-\frac{r}{2}=\frac{2 r}{3}-\frac{7}{2}$

$\Rightarrow \frac{2 r}{3}+\frac{r}{2}=7+\frac{7}{2}$

$\Rightarrow \frac{7 r}{6}=\frac{21}{2}$

$\Rightarrow r=9$

Hence, the required term is the 10 th term