Points of intersections

Probability Level 4

What are the (maximum) number of points of intersection of diagonals in the interior of a 50-sided polygon.

Try my set : Let's play with polygons


The answer is 230300.

Aniket Sanghi by Aniket Sanghi , 21, India

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2 solutions

Aniket Sanghi
Apr 2, 2016

Select any 4 vertices, these will form a quadrilateral whose diagonals intersect max at 1 point.

Hence total no. Of such quadrilaterals is 50C4 = 230300

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it's an example from Arthur Engel

Rajdeep Ghosh - 3 years, 2 months ago

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Oh! Gr8! Though I never read that book!

Aniket Sanghi - 3 years, 2 months ago
Vijay Simha
Mar 20, 2018

The (maximum) number of points of intersection of diagonals in the interior of a n-sided polygon is n (n -1) (n - 2)*(n - 3)/24.

For a square it is 4 * 3 * 2 * 1/24 = 1

For a Pentagon it is 5 * 4 * 3 * 2/24 = 5

For a Hexagon it is 6 * 5 *4 *3 /24 = 15 -> ( You need to consider that three pairs of diagonals shared a single point in the center as their intersection)

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