Phương pháp giải:

\(A_n^k = \frac{{n!}}{{\left( {n - k} \right)!}},\,\,C_n^k = \frac{{n!}}{{k!\left( {n - k} \right)!}}\).

Lời giải chi tiết:

Điều kiện: \(x \in N,\,\,x \ge 3\).

\(\begin{array}{l}A_x^3 + C_x^{x - 3} = 14x \Leftrightarrow \frac{{x!}}{{\left( {x - 3} \right)!}} + \frac{{x!}}{{\left( {x - 3} \right)!3!}} = 14x \Leftrightarrow \frac{7}{6}.\frac{{x!}}{{\left( {x - 3} \right)!}} = 14x \Leftrightarrow \frac{7}{6}x\left( {x - 1} \right)\left( {x - 2} \right) = 14x\\ \Leftrightarrow x\left( {x - 1} \right)\left( {x - 2} \right) - 12x = 0 \Leftrightarrow x\left( {{x^2} - 3x - 10} \right) = 0 \Leftrightarrow \left[ \begin{array}{l}x = 0\,\,(L)\\x =  - 2\,\,(L)\\x = 5\,\,(TM)\end{array} \right. \Leftrightarrow x = 5\end{array}\)

Chọn: A