Number of regular polygons possible 
Thirty six equally spaced points are plotted on a circle, and some of
these points are joined successively to form a polygon. How many
distinct such regular polygons are possible.

What I thought that the answer should be $^{36}C_3 + ^{36}C_4 + ^{36}C_5 + ^{36}C_6 + ^{36}C_7 + \cdots + ^{36}C_{36}$ but this is not the correct answer that has been provided. What am I doing wrong? Please help !!!
Thanks in advance !!!

Since the $k$ vertices of a regular polygon inscribed in a circle must be equally spaced, to form a regular polygon using the $36$ equally spaced points on the circle, the number of sides must be a divisor of $36$.  Hence, the regular polygon must have either $3, 4, 6, 9, 12, 18,$ or $36$ sides.  There are $36/k$ distinguishable regular polygons with $k$ sides since there are $36$ possible points at which you could start drawing the regular polygon, but doing so counts each such regular polygon $k$ times, once for each of the $k$ vertices of the polygon where you could start your drawing.  Hence, the number of distinct regular polygons which could be drawn using $36$ equally spaced points on the circle is
$$\frac{36}{3} + \frac{36}{4} + \frac{36}{6} + \frac{36}{9} + \frac{36}{12} + \frac{36}{18} + \frac{36}{36} = 12 + 9 + 6 + 4 + 3 + 2 + 1 = 37$$
