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. $\frac{20!}{15!2!3!}$
• D. none of these

Answer 1

Answer: (C)

$\frac{20!}{15!2!3!}$

Coefficient of $x^2{y}^3$ in ${(1-x+y)}^{20}$

Sum of powers of $x$ and $y$$=5$

${}_{}^{20}{C}_5{1}^5{(-x+y)}^5$

${}^{20}{C}_5{(-1)}^5{\times }^5{C}_3{(-x)}^2{y}^3$

${}^{20}{C}_5(-1){\times }^5{C}_3{\left(x\right)}^2{y}^3$

$=\frac{-20!}{5!15!}\times \frac{5!}{3!2!}=\frac{-20!}{15!3!2!}$