Question

The coefficient of $x^6.y^{-2}$ in the in the expansion of ${\left(\frac{x^2}{y}-\frac{y}{x}\right)}^{12}$ is

• A. ${}^{12}{\text{C}}_6$
• B. ${-}^{12}{\text{C}}_5$
• C. $0$
• D. none of these

Answer 1

The correct option is C $0$

The general term for the above expression is
$T_{r+1}=(-1{)}^r{}^{12}{C}_r{x}^{24-3r}{y}^{2r-12}$
For the term $x^6{y}^{-2}$
$24-3r=6$
$18=3r$
$r=6$
And $2r^{\prime }-12=-2$
$2r^{\prime }=10$
$r^{\prime }=5$
Since r and r' do not coincide, the following term does not exists.