Number of Triangles in a Regular Polygon: Is nC3 the Correct Formula?

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SUMMARY

The correct formula for determining the number of triangles that can be formed by the vertices of a regular polygon with n sides is indeed nC3. This conclusion is reached by selecting any three vertices from the n available vertices, which guarantees the formation of a triangle. To verify this, one can draw various regular polygons and count the triangles formed, ensuring that the conditions of vertex selection and triangle uniqueness are met.

PREREQUISITES
  • Understanding of combinatorial mathematics, specifically combinations.
  • Familiarity with the concept of vertices in geometry.
  • Basic knowledge of regular polygons and their properties.
  • Ability to visualize geometric shapes and their configurations.
NEXT STEPS
  • Study the principles of combinatorial mathematics, focusing on the binomial coefficient nCk.
  • Explore geometric properties of regular polygons, including vertex arrangements.
  • Learn about the uniqueness of triangles formed by different vertex selections in polygons.
  • Practice problems involving the counting of shapes formed by vertices in various geometric configurations.
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Students studying combinatorial geometry, mathematics educators, and anyone interested in understanding the properties of regular polygons and triangle formation.

utkarshakash
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Homework Statement


Find number of triangles which can be obtained by vertices of a regular polygon of n sides.

The Attempt at a Solution


I think it should be nC3 as forming a triangle requires you to select any 3 vertices from n available vertices. But I'm not sure whether this is correct or not. Can anyone verify it?
 
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I think the way you would go about verifying something like that would be to draw a bunch of regular polygons, then work out the triangles - count them, and see if it agrees with nC3.
You'll at least gain confidence with each one that works - but the trick is to see if you can figure out how to get one that won't.
 
You just have to satisfy yourself in regard to four questions:
1. For any such triangle, is there a selection of three vertices that produces it?
2. Does every such selection produce a triangle?
3. Does every such selection produce only one triangle?
4. For any two such selections, do they produce different triangles?
 

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