Solving Equation in Binomial Coefficients Example

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
$n=3, \: 8$