Giải hệ phương trình \(\left\{\begin{array}{l} \frac{\mathrm{C}_{n+1}^{m+1}}{\mathrm{C}_{n+1}^{m}}=1 \\ \frac{\mathrm{C}_{n+1}^{m}}{\mathrm{C}_{n+1}^{m-1}}=\frac{5}{3} \end{array}\right.\)

\(\text { Điều kiện } m, n \in \mathbb{N}^{*}, m \leq n \text { . }\)

Ta có:

\(\frac{\mathrm{C}_{n+1}^{m+1}}{\mathrm{C}_{n+1}^{m}}=\frac{\frac{(n+1) !}{(m+1) !(n-m) !}}{\frac{(n+1) !}{m !(n+1-m) !}}=\frac{n+1-m}{m+1} ;\)

\(\frac{\mathrm{C}_{n+1}^{m}}{\mathrm{C}_{n+1}^{m-1}}=\frac{\frac{(n+1) !}{m !(n+1-m) !}}{\frac{(n+1) !}{(m-1) !(n+2-m) !}}=\frac{n+2-m}{m} .\)

\(\text { Suy ra }\left\{\begin{array} { l }
{ \frac { n + 1 – m } { m + 1 } = 1 } \\
{ \frac { n + 2 – m } { m } = \frac { 5 } { 3 } }
\end{array} \Leftrightarrow \left\{\begin{array} { l }
{ n = 2 m } \\
{ 3 n – 8 m + 6 = 0 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
m=3 \\
n=6
\end{array}\right.\right.\right. \text { (thỏa mãn). }\)Vậy m = 3, n = 6 là các giá trị cần tìm.