Question

The coefficient of $x^2{y}^3$ in the expansion of $(1-x+y{)}^{20}$ is

• A. $\frac{20!}{2!3!}$
• B. $-\frac{20!}{2!3!}$
• C. $\frac{20!}{5!2!3!}$
• D. $\frac{20!}{(15!)(3!)(2!)}$

Answer 1

The correct option is D $\frac{20!}{(15!)(3!)(2!)}$

The general term in the expansion of $(1-x+y{)}^{20}$
$=\frac{20!}{k_1!k_2!k_3!}{1}^{k_1}(-x{)}^{k_2}(y{)}^{k_3}$

Where, $k_1+k_2+k_3=20$
For $x^2{y}^3$, so
$k_2=2,k_3=3\Rightarrow k_1=15$

Hence, the coefficient of $x^2{y}^3$
$=\frac{20!}{(15!)(3!)(2!)}$