2021 Fall AMC 12B Problem 19

Regular polygons with 5,6,7, and 8 sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?

(\textbf{A})\: 52\qquad(\textbf{B}) \: 56\qquad(\textbf{C}) \: 60\qquad(\textbf{D}) \: 64\qquad(\textbf{E}) \: 68

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.