jmc

With the ten points on the circumference of a circle, any set of 4 of them will form a convex (indeed, cyclic) quadrilateral. So, with ten points, and we can choose any 4 of them to form a distinct quadrilateral, we get $\left(\genfrac{}{}{0ex}{}{10}{4}\right)=\frac{10\cdot 9\cdot 8\cdot 7}{4\cdot 3\cdot 2}=10\cdot 3\cdot 7=\boxed{210}$ quadrilaterals.