Number of r-polygons in an n-polygon with no side coinciding

Here is the full question:

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.

I imagined $(n-r)$ vertices in a closed polygon. There are $(n-r)$ possibilities for adding r vertices between them. (If we add r vertices here then no 2 vertices will be together). This leads me to $(}\genfrac{}{}{0ex}{}{n-r}{r}{\textstyle )$. But the correct answer wants me to multiply it with $\frac{n}{n-r}$. What is the need for the last step?

Here is the full question:

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.

I imagined $(n-r)$ vertices in a closed polygon. There are $(n-r)$ possibilities for adding r vertices between them. (If we add r vertices here then no 2 vertices will be together). This leads me to $(}\genfrac{}{}{0ex}{}{n-r}{r}{\textstyle )$. But the correct answer wants me to multiply it with $\frac{n}{n-r}$. What is the need for the last step?