\(\displaystyle{\frac{{{n}!}}{{{\left({n}-{3}\right)}!}}}\)

\(\displaystyle={\frac{{{n}{\left({n}-{1}\right)}{\left({n}-{2}\right)}{\left({n}-{3}\right)}{\left({n}-{4}\right)}\ldots.\times{3}\times{2}\times{1}}}{{{\left({n}-{3}\right)}{\left({n}-{4}\right)}\ldots\times{3}\times{2}\times{1}}}}\)

\(\displaystyle={n}{\left({n}-{1}\right)}{\left({n}-{2}\right)}\)

\(\displaystyle{n}{\left({n}^{{{2}}}-{3}{n}+{2}\right)}\)

\(\displaystyle={n}^{{{2}}}-{3}{n}^{{{2}}}+{2}{n}\)

\(\displaystyle={\frac{{{n}{\left({n}-{1}\right)}{\left({n}-{2}\right)}{\left({n}-{3}\right)}{\left({n}-{4}\right)}\ldots.\times{3}\times{2}\times{1}}}{{{\left({n}-{3}\right)}{\left({n}-{4}\right)}\ldots\times{3}\times{2}\times{1}}}}\)

\(\displaystyle={n}{\left({n}-{1}\right)}{\left({n}-{2}\right)}\)

\(\displaystyle{n}{\left({n}^{{{2}}}-{3}{n}+{2}\right)}\)

\(\displaystyle={n}^{{{2}}}-{3}{n}^{{{2}}}+{2}{n}\)