Criss-Cross

Probability Level 3

Eighteen points are chosen on a circle. An octadecagon ( $18$ sided polygon) is drawn. All of its vertices are joined to each other by straight lines (the polygon is designed in such a way that no three of these lines will be concurrent). Obviously, these straight lines will intersect each other, either inside or outside the polygon.

Find the total number of internal intersection points. (Vertices are not internal)

The answer is 3060.

2 solutions

David Vreken
Dec 22, 2018

The total number of internal intersection points is the same as the number of times $2$ different diagonals intersect, and since $2$ different diagonals require $4$ different vertices.

Therefore, the total is equivalent to the number of ways of choosing $4$ vettices out of $18$ , which is ${18 \choose 4} = \boxed{3060}$ .

Tolga Gürol
Dec 22, 2018

Eighteen points are chosen on a circle. An octadecagon ( $18$ sided polygon) is drawn. All of its vertices are joined to each other by straight lines (the polygon is designed in such a way that no three of these lines will be concurrent and these straight lines will intersect each other, either inside or outside the polygon)

Find the total number of internal and external intersection points. (Vertices are neither internal nor external)

Question should be asked like that if the answer is $3060$

Number of internal intersections is 3060, not total. External intersections would give more points. I'm taking about straight lines that would join the vertices, there are possibilities where these lines will meet outside the polygon/circle. For example, any two non adjacent sides of a pentagon when produced meet at a point outside the pentagon.

Parth Sankhe - 2 years, 5 months ago

Now I see that it is my mistake. I found the answer accidentally.

Tolga Gürol - 2 years, 5 months ago

Criss-Cross

The answer is 3060.

2 solutions

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