2017 UNCO Math Contest II Problems/Problem 9

Problem

$[asy] pair A=dir(0),B=dir(80),C=dir(160),D=dir(310),O=(0,0); draw(circle(O,1),black); draw(A--B--C--D--A--C--B--D); [/asy]$

Suppose n points on the circumference of a circle are joined by straight line segments in all possible ways and that no point that is not one of the original n points is contained in more than two of the segments. How many triangles are formed by the segments? Count all triangles whose sides lie along the segments, including triangles that overlap with other triangles. For example, for n = 3 there is one triangle and for n = 4 (shown in the diagram) there are 8 triangles.

Solution

${n}\choose{6}$ + 5 ${n}\choose{5}$ + 4 ${n}\choose{4}$ + ${n}\choose{3}$

2017 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 9
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2017 UNCO Math Contest II Problems/Problem 9

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