Lời giải chi tiết:

\(C_n^3 + 3A_n^2 = 390\)\(\left( {n \ge 3;\,\,n \in \mathbb{N}} \right)\)

\( \Leftrightarrow \dfrac{{n!}}{{3!(n - 3)!}} + 3\dfrac{{n!}}{{\left( {n - 2} \right)!}} = 390\)

\( \Leftrightarrow \dfrac{1}{{3!}}n\left( {n - 1} \right)\left( {n - 2} \right) + 3n\left( {n - 1} \right) = 390\)

\( \Leftrightarrow \dfrac{1}{6}\left( {{n^2} - n} \right)\left( {n - 2} \right) + 3{n^2} - 3n - 390 = 0\)

\( \Leftrightarrow \dfrac{1}{6}{n^3} - \dfrac{1}{2}{n^2} + \dfrac{1}{3}n + 3{n^2} - 3n - 390 = 0\)

\( \Leftrightarrow \dfrac{1}{6}{n^3} + \dfrac{5}{2}{n^2} - \dfrac{8}{3}n - 390 = 0\)

\( \Leftrightarrow n = 10\)

Chọn C.