prove the following :

$\mathrm{given} \alpha =\mathrm{binomial\_coefficient}(n, 2)=\frac{n(n-1)}{2!}=\frac{n(n-1)}{2}$$\mathrm{Now} \mathrm{binomial\_coefficient}(\alpha , 2)=\frac{\alpha (\alpha -1)}{2!}=\frac{\alpha (\alpha -1)}{2}=\frac{1}{2}\frac{n(n-1)}{2}\left[\frac{n(n-1)}{2}-1\right]$$=\frac{1}{2}\frac{n(n-1)}{2}\left[\frac{n^2-n-2}{2}\right]$$= \frac{1}{2}\frac{n(n-1)}{2}\left[\frac{(n-2)(n+1)}{2}\right]$$=\frac{(n+1)n(n-1)(n-2)}{8}$$\mathrm{Now} 3\mathrm{binomial\_coefficient}(n+1, 4)= 3\frac{(n+1)!}{(n-3)!4!}=\frac{(n+1)n(n-1)(n-2)}{8}$$\mathrm{Hence} \mathrm{proved}.$">