Question

Rút gọn tổng \(S=C\overset{1}{2019}-2C\overset{2}{2019}+...-2018C\overset{2018}{2019}+2019C\overset{2019}{2019}\) bằng:

A. 2019

B.1

C. -2019

D. 0

Answer 1

Xét khai triển:

\(\left(1-x\right)^{2019}=C_{2019}^0-C_{2019}^1x+C_{2019}^2x^2-...+C_{2019}^{2018}x^{2018}-C_{2019}^{2019}x^{2019}\)

Đạo hàm 2 vế:

\(\Rightarrow-2019\left(1-x\right)^{2018}=-C_{2019}^1+2C_{2019}^2x-...+2018C_{2019}^{2018}x^{2017}-2019C_{2019}^{2019}x^{2018}\)

Thay \(x=1\)

\(\Rightarrow-C_{2019}^1+2C_{2019}^2-....+2018C_{2019}^{2018}-2019C_{2019}^{2019}=0\)

\(\Rightarrow-S=0\Rightarrow S=0\)