Question

Find the coefficient of $x^2{y}^5$ in the expansion of ${(3+2x-y)}^{10}$

• A. ${}^{10}{C}_5{\times }^5{C}_2\times 108$
• B. ${}^{10}{C}_2{\times }^5{C}_2\times 108$
• C. ${}^{-10}{C}_2{\times }^5{C}_2\times 108$
• D. ${}^{-10}{C}_5{\times }^5{C}_2\times 108$

Answer 1

The correct option is D ${}^{-10}{C}_5{\times }^5{C}_2\times 108$

$x^2{y}^5$ can be obtained as
$\left(3{)}^3\right(2x{)}^2(-y{)}^5$
$=-3^3{.2}^2{x}^2{y}^5$
Hence the required coefficient is
$=-\frac{10!}{3!.2!.5!}.4.(3{)}^3$
$=-\frac{10!}{5!.5!}.\frac{5!}{3!.2!}.108$
$=-{}^{10}{C}_5.{}^5{C}_{2}108$