Problem of the Month - September 2023

Seven points are randomly placed on the circle shown below. 

How many different triangles can be drawn using these points as vertices?

A circle with seven distinct points on its edge


There are 35 possible triangles.  If we label the points A, B, C, D, E, F, and G then the triangles are: ABC, ABD, ABE, ABF, ABG, ACD, ACE, ACF, ACG, ADE, ADF, ADG, AEF, AEG, AFG, BCD, BCE, BCF, BCG, BDE, BDF, BDG, BEF, BEG, BFG, CDE, CDF, CDG, CEF, CEG, CFG, DEF, DEG, DFG, and EFG.  

How many quadrilaterals are possible?  Would there be more quadrilaterals or more triangles?  Is there a quick way to figure out how many quadrilaterals there would be?