2021 Fall AMC 10B Problem 21

For a regular $k$-gon and a regular $n$-gon inscribed in the same circle with $k\lt n$ and no shared vertices, their boundaries intersect in $2k$ points. Each side of the smaller polygon is crossed twice by the boundary of the larger polygon. Therefore, sum over all pairs of polygons. The $5$-gon contributes $2\cdot5$ intersections with each of the $6$-, $7$-, and $8$-gons. The $6$-gon contributes $2\cdot6$ intersections with each of the $7$- and $8$-gons. The $7$-gon contributes $2\cdot7$ with the $8$-gon. The total is $$3(10)+2(12)+14=68.$$ Thus, the answer is E .