Lời giải chi tiết:

\(A_n^3 = 20n\)\(\left( {n \ge 3;\,\,n \in \mathbb{N}} \right)\)

\( \Leftrightarrow \dfrac{{n!}}{{\left( {n - 3} \right)!}} = 20n\)

\( \Leftrightarrow n\left( {n - 1} \right)\left( {n - 2} \right) - 20n = 0\)

\( \Leftrightarrow {n^3} - 3{n^2} - 18n = 0\)

\( \Leftrightarrow \left[ \begin{array}{l}n =  - 3\,\,\,\,\left( {ktm} \right)\\n = 0\,\,\,\,\,\,\,\left( {ktm} \right)\\n = 6\,\,\,\,\,\,\,\,\,\left( {tm} \right)\end{array} \right.\)

Chọn A.