# Which Term in the Expansion of ⎧ ⎨ ⎩ ( X √ Y ) 1 / 3 + ( Y X 1 / 3 ) 1 / 2 ⎫ ⎬ ⎭ 21 Contains X and Y to One and the Same Power? - Mathematics

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 Tr+1th term in the given expression contains x and y to one and the same power.
Then,  $T_{r + 1} \text{ th term is}$
$^{21}{}{C}_r \left[ \left( \frac{x}{\sqrt{y}} \right)^{1/3} \right]^{21 - r} \left[ \left( \frac{y}{x^{1/3}} \right)^{{}^{1/2}} \right]^r$
$=^{21}{}{C}_r \left( \frac{x^{(21 - r)/3}}{x^{r/6}} \right)\left( \frac{y^{r/2}}{y^{(21 - r)/6}} \right)$
$= ^{21}{}{C}_r \left( x \right)^{7 - r/2} \left( y \right)^{2r/3 - 7/2}$
$\text{ Now, if x and y have the same power, then }$
$7 - \frac{r}{2} = \frac{2r}{3} - \frac{7}{2}$
$\Rightarrow \frac{2r}{3} + \frac{r}{2} = 7 + \frac{7}{2}$
$\Rightarrow \frac{7r}{6} = \frac{21}{2}$
$\Rightarrow r = 9$
$\text{ Hence, the required term is the 10th term }$

Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 18 Binomial Theorem
Exercise 18.2 | Q 10 | Page 38