The coefficient of `x^2y^3` in the expansion of `(1-x+y)^(20)` is `(20 !)/(213 !)` b. `-(20 !)/(213 !)` c. `(20 !)/(15!2!3!)` d. none of these

Options

A. `(20!)/(2!3!)`
B. `-(20!)/(2!3!)`
C. `(20!)/(5!2!3!)`
D. `(20!)/(15!2!3!)`

Answer

D

Explanation

The general term in the expansion of `(1-x+y)^(20)` is
`(20!)/(r!s!t!)1^(r)(-x)^(s)(y)^(t)`, where `r+s+t=20`
For `x^(2)y^(3)`, we have the term
`(20!)/(15!2!3!) 1^(15)(-x)^(2)(y)^(3)`
Hence, the coefficient of `x^(2)y^(3)` is
`(20!)/(15!2!3!)`