What is the coefficient of `x^(5)` in the expansion `(1-2x+3x^(2)-4x^(3)+……oo)^(-5)` ?

Options

A. `(10!)//(5!)^(2)`
B. `5^(-5)`
C. `5^(5)`
D. `10!//{6!)(4!)}`

Answer

A

Explanation

`1-2x+3x^(2)-4x^(3)+…..`
`=(1+x)^(2), so (1-2x+3x^(2)-4x^(3)+….oo)^(-5)`
`=((1+x^(-2))=(1+x)^(10) rArr T_(r+1)=""^(10)C_(r)x^®`
Putting r=5, coefficient of `x^(5)=""^(10)C_(5)=(10!)/(5!5!)=(10!)/((5!)^(2))`