Question

The middle term in the expansion of ${(\frac{2}{3}x-\frac{3}{2}y)}^{20}$ is

• A. ${}^{20}{C}_{10}{x}^{10}{y}^{10}$
• B. $x^{10}{y}^{10}$
• C. ${}^{20}{C}_{10}{\left(2/3\right)}^{10}{\left(xy\right)}^{10}$
• D. $-x^{10}{y}^{10}$

Answer 1

The correct option is A ${}^{20}{C}_{10}{x}^{10}{y}^{10}$

${(\frac{2}{3}x-\frac{3}{2}y)}^{20}$

$\text{There will be total}21\text{terms in this exapnsion}$

$\text{: :}11\text{th term will be middle tem.}$

$T_{r+1=}{}^n{C}_r\cdot x^{n-r}\cdot y^r$

$\Rightarrow T_{11}={}^{20}{c}_{10}\cdot {\left(\frac{2}{3}x\right)}^{20-10}{\left(\frac{-3}{2}y\right)}^{10}$

$={}^{20}{c}_{10}{x}^{10}\cdot y^{10}$

option (a)