Cho tổng \(S=C_{2017}^{1}+C_{2017}^{2}+....+C_{2017}^{2017}\) Giá trị tổng S bằng: A.\({{2}^{2018}}\) B.\({{2}^{2017}}\) C. \({{2}^{2017}}-1\) D. \({{2}^{2016}}\)
Đáp án đúng: C Giải chi tiết:Xét khai triển: \({{\left( x+1 \right)}^{2017}}=C_{2017}^{0}{{x}^{2017}}+C_{2017}^{1}{{x}^{2016}}+C_{2017}^{2}{{x}^{2015}}+........+C_{2017}^{2016}x+C_{2017}^{2017}.\) Với \(x=1\) ta có: \({{\left( 1+1 \right)}^{2017}}=C_{2017}^{0}{{.1}^{2017}}+C_{2017}^{1}{{.1}^{2016}}+C_{2017}^{2}{{.1}^{2015}}+........+C_{2017}^{2016}.1+C_{2017}^{2017}\) \(\begin{align} & \Leftrightarrow {{2}^{2017}}=C_{2017}^{0}+C_{2017}^{1}+C_{2017}^{2}+........+C_{2017}^{2016}+C_{2017}^{2017} \\ & \Leftrightarrow {{2}^{2017}}-C_{2017}^{0}=C_{2017}^{1}+C_{2017}^{2}+........+C_{2017}^{2016}+C_{2017}^{2017} \\ & \Leftrightarrow S={{2}^{2017}}-C_{2017}^{0}={{2}^{2017}}-1. \\\end{align}\) Chọn C.