To solve the equations
\[\begin{pmatrix}n\\3\end{pmatrix} =3 \begin{pmatrix}n-1\\2\end{pmatrix} - \begin{pmatrix}n-1\\1\end{pmatrix}\]
evaluate each binomial coefficient in terms of \[n\]
.\[\frac{n(n-1)(n-2)}{6}=3 \frac{(n-1)(n-2)}{2}- (n-1)\]
\[\frac{n(n-2)}{6}=3 \frac{n-2}{2}- 1\]
\[n(n-2)=9(n-2)- 6\]
\[n^2-2n=9n-24\]
\[n^2-11n+24=0\]
\[(n-3)(n-8)=0\]
Hence
\[n=3, \: 8\]