Lời giải chi tiết:

\(S = C_n^0 - 3C_n^1 + {3^2}C_n^2 - {3^3}C_n^3 + ... + {\left( { - 1} \right)^n}{3^n}C_n^n\)

Xét khai triển: \({\left( {1 - x} \right)^n} = C_n^0{.1^n} + C_n^1{.1^{n - 1}}.\left( { - x} \right) + C_n^2{.1^{n - 2}}.{\left( { - x} \right)^2} + ... + C_n^{n - 1}.1.{\left( { - x} \right)^{n - 1}} + C_n^n.{\left( { - x} \right)^n}\)

Chọn n= 3  ta được:

\({\left( {1 - 3} \right)^n} = C_n^0 - 3C_n^1 + {3^2}C_n^2 + ... + C_n^{n - 1}.{\left( { - 3} \right)^{n - 1}} + C_n^n.{\left( { - 3} \right)^n}\)

\( = C_n^0 - 3C_n^1 + {3^2}C_n^2 + ... + C_n^n{\left( { - 1} \right)^n}{.3^n} = {\left( { - 2} \right)^n}\)

Chọn D.