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

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Jim Hunt
$$\displaystyle{\frac{{{n}!}}{{{\left({n}-{3}\right)}!}}}$$
$$\displaystyle={\frac{{{n}{\left({n}-{1}\right)}{\left({n}-{2}\right)}{\left({n}-{3}\right)}{\left({n}-{4}\right)}\ldots.\times{3}\times{2}\times{1}}}{{{\left({n}-{3}\right)}{\left({n}-{4}\right)}\ldots\times{3}\times{2}\times{1}}}}$$
$$\displaystyle={n}{\left({n}-{1}\right)}{\left({n}-{2}\right)}$$
$$\displaystyle{n}{\left({n}^{{{2}}}-{3}{n}+{2}\right)}$$
$$\displaystyle={n}^{{{2}}}-{3}{n}^{{{2}}}+{2}{n}$$
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Paineow
$$\displaystyle{\frac{{{n}!}}{{{\left({n}-{3}\right)}!}}}$$
Cancel the factorials: $$\displaystyle{\frac{{{n}!}}{{{\left({n}-{m}\right)}!}}}={n}\times{\left({n}-{1}\right)}\ldots{\left({n}-{m}+{1}\right)}$$, n>m
Thus, we have
$$\displaystyle{\frac{{{n}!}}{{{\left({n}-{3}\right)}!}}}={n}{\left({n}-{1}\right)}{\left({n}-{2}\right)}$$
RizerMix

$$\frac{n!}{(n-3)!}=n(n-1)(n-2)=n^{2}-3n^{2}+2n$$