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Chapter 1: Problem 21

How many triangles are determined by the vertices of a regular polygon of \(n\) sides? How many if no side of the polygon is to be a side of any triangle?

Short Answer

Expert verified
The total number of triangles that can be formed by the vertices of a regular polygon with \(n\) sides is \(\frac{n(n-1)(n-2)}{3!}\). If no side of the polygon is to be a side of any triangle, the number reduces to \(\frac{n(n-1)(n-2)}{3!} - n\).

Step by step solution

01

Calculate Total Triangles

To find the total number of triangles that can be formed using the vertices of a regular polygon with \(n\) sides, apply the combination formula. This is because a triangle is formed by selecting 3 points out of \(n\). The number of different combinations of \(n\) elements taken 3 at a time is represented as \(C(n, 3)\). Hence, the total number of triangles is \(C(n, 3) = \frac{n(n-1)(n-2)}{3!}\).
02

Calculate Triangles Excluding Those with Sides on the Polygon

Triangles that have their sides on the polygon are those formed by adjacent vertices. Since there are \(n\) vertices and each vertex is adjacent to 2 others, there are a total of \(n\) such triangles. So, subtract the number of these triangles from the total number of triangles calculated in Step 1. Hence, the number of triangles without sides on the polygon is \(C(n, 3) - n = \frac{n(n-1)(n-2)}{3!} - n\).

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