Xét khai triển:
\(\left(1+x\right)^{2019}=C_{2019}^0+xC_{2019}^1+...+x^{2019}C_{2019}^{2019}\)
Đạo hàm 2 vế:
\(2019\left(1+x\right)^{2018}=C_{2019}^1+2xC_{2019}^2+...+2019x^{2018}C_{2019}^{2019}\)
\(\Rightarrow2019x\left(1+x\right)^{2018}=xC_{2019}^1+2x^2C_{2019}^2+3x^3C_{2019}^3+...+2019x^{2019}C_{2019}^{2019}\)
Đạo hàm 2 vế:
\(2019\left(1+x\right)^{2018}+2018.2019x\left(1+x\right)^{2017}=C_{2019}^1+2^2xC_{2019}^2+...+2019^2x^{2018}C_{2019}^{2019}\)
Thay \(x=1\)
\(\Rightarrow2019.2^{2018}+2018.2019.2^{2017}=C_{2019}^1+2^2C_{2019}^2+...+2019^2C_{2019}^{2019}\)
\(\Rightarrow S=2019.2^{2018}+2018.2019.2^{2017}=2019.2^{2018}.\left(1+1009\right)=2019.505.2^{2019}\)
