r-sided polygons are formed by joining the vertices of an n-sided polygon. Find the number of polygons that can be formed, none of whose sides coincide with those of the n-sided polygon.
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$$\begin{gathered} \mathrm{We} \mathrm{can} \mathrm{do} \mathrm{it} \mathrm{by} \mathrm{stars} \mathrm{and} \mathrm{bars} \mathrm{analogy} \left(\mathrm{Read} \mathrm{wiki} \mathrm{of} \mathrm{stars} \mathrm{and} \mathrm{bars}\right) \\ \mathrm{If} \mathrm{we} \mathrm{represent} \mathrm{a} \mathrm{used} \mathrm{vertex} \mathrm{as} \bullet \mathrm{and} \mathrm{unused} \mathrm{vertex} \mathrm{as} , \mathrm{then} \mathrm{one} \mathrm{of} \mathrm{the} \\ \mathrm{possible} \mathrm{polygon} \mathrm{will} \mathrm{look} \mathrm{like} \\ ○\bullet ○○\bullet ○\bullet ○○○\bullet ○○ \\ \left(\mathrm{For} \mathrm{above} \mathrm{example} \mathrm{n}=14 \mathrm{and} \mathrm{r}=4\right). \\ \mathrm{If} \mathrm{the} \mathrm{last} \mathrm{vertex} \mathrm{is} \mathrm{unused}, \mathrm{then} \mathrm{we} \mathrm{can} \mathrm{assume} \mathrm{of} \mathrm{this} \mathrm{as} \mathrm{an} \mathrm{arbitrary} \\ \mathrm{string} \mathrm{of} \mathrm{n}-2\mathrm{r} \text{'}○\text{'} \mathrm{s} \mathrm{and} \mathrm{r} \text{'}\bullet \text{'}\text{'}\mathrm{s}- \mathrm{there} \mathrm{are}{ }^{\mathrm{n}-\mathrm{r}}{\mathrm{C}}_{\mathrm{r}} \mathrm{such} \mathrm{strings} \mathrm{and} \mathrm{of} \mathrm{total} \\ \mathrm{polygons} \frac{\mathrm{n}-\mathrm{r}}{\mathrm{n}} \mathrm{fractions} \mathrm{do} \mathrm{not} \mathrm{use} \mathrm{last} \mathrm{vertex}, \mathrm{hence} \\ \mathrm{Number} \mathrm{of} \mathrm{required} \mathrm{polygons}=\left(\frac{\mathrm{n}}{\mathrm{n}-\mathrm{r}}\right){ }^{\mathrm{n}-\mathrm{r}}{\mathrm{C}}_{\mathrm{r}} \\ =\left(\frac{\mathrm{n}}{\mathrm{n}-\mathrm{r}}\right) \frac{\left(\mathrm{n}-\mathrm{r}\right)!}{\mathrm{r}! \left(\mathrm{n}-\mathrm{r}-\mathrm{r}\right)!} \\ =\left(\frac{\mathrm{n}}{\mathrm{n}-\mathrm{r}}\right) \frac{\left(\mathrm{n}-\mathrm{r}\right)\left(\mathrm{n}-\mathrm{r}-1\right)!}{\mathrm{r}! \left(\mathrm{n}-2\mathrm{r}\right)!} \\ =\frac{\mathbf{n}\left(\mathbf{n}\mathbf{-}\mathbf{r}\mathbf{-}\mathbf{1}\right)\mathbf{!}}{\mathbf{r}\mathbf{!}\mathbf{ }\left(\mathbf{n}\mathbf{-}\mathbf{2}\mathbf{r}\right)\mathbf{!}}\mathbf{ }\left(\mathbf{Answer}\right) \end{gathered}$$

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