Lời giải chi tiết:

A. \(C_n^k = \frac{{n!}}{{\left( {n - k} \right)!.k!}};{\rm{       }}\frac{n}{k}C_{n - 1}^{k - 1} = \frac{{n\left( {n - 1} \right)!}}{{k\left( {n - k} \right)!\left( {k - 1} \right)!}} = \frac{{n!}}{{\left( {n - k} \right)!k!}}\)

B. \(C_n^{k + 1} = \frac{{n!}}{{\left( {n - k - 1} \right)!\left( {k + 1} \right)!}};{\rm{       }}C_n^{n - k - 1} = \frac{{n!}}{{\left( {k + 1} \right)!\left( {n - k - 1} \right)!}}\)

C. \(C_{n + 1}^k = \frac{{\left( {n + 1} \right)!}}{{\left( {n + 1 - k} \right)!k!}};{\rm{        }}C_n^k + C_n^{k - 1} = \frac{{n!}}{{\left( {n - k} \right)!k!}} + \frac{{n!}}{{\left( {n - k + 1} \right)!\left( {k - 1} \right)!}}\)

 \( = \frac{{n!\left( {n - k + 1} \right)}}{{\left( {n - k + 1} \right)!k!}} + \frac{{n!k}}{{\left( {n - k + 1} \right)!k!}} = n!\left[ {\frac{{n + 1}}{{\left( {n - k + 1} \right)!k!}}} \right] = \frac{{\left( {n + 1} \right)!}}{{\left( {n - k + 1} \right)!k!}}\)

D. \(C_{n + 1}^{k + 1} = \frac{{\left( {n + 1} \right)!}}{{\left( {n - k} \right)!\left( {k + 1} \right)!}};{\rm{      }}C_{n + 1}^{n - k - 1} = \frac{{\left( {n + 1} \right)!}}{{\left( {k + 2} \right)!\left( {n - k - 1} \right)!}}\)

Chọn D.