Question

There are six points in a plane, no three of which are collinear, In how many ways can you draw a pair of triangles with the six points as vertices.

Short answer

There are 190 ways to choose a pair of triangles from six points.

Step-by-step solution

Determine Total Combinations of Triangles

Since no three points are collinear, any selection of three points forms a triangle. We first compute the total number of ways to choose 3 points out of 6. This is given by the combination formula \( \binom{n}{r} \) where \( n \) is the total number of items and \( r \) is the number of items to choose. Thus, the total number of triangles is:\[ \binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20. \]

Calculate Ways to Choose Two Triangles

To find the number of ways to choose a pair of triangles from the 20 possible triangles, we need to select 2 triangles out of these 20. Again, we use the combination formula:\[ \binom{20}{2} = \frac{20 \times 19}{2 \times 1} = 190. \]

Key concepts

Understanding Triangles in Combinatorics

In combinatorics, a triangle is formed by selecting any three points that do not lie on a single line. This is why we say that no three points should be collinear. Triangles are fundamental geometric shapes, and in problems involving combinations, determining the number of possible triangles is a common task.

For example, if you have six distinct points in a plane and none of them are collinear, you can form a wide variety of triangles. The key here is to realize that each set of three points will yield a unique triangle. This understanding is crucial when calculating combinations in geometric contexts, such as the exercise provided.

The Combination Formula Explained

The combination formula is a mathematical tool used to determine how many ways you can select items from a larger set. It is commonly denoted as \( \binom{n}{r} \), pronounced "n choose r," where \( n \) represents the total items to choose from, and \( r \) is the number of items we want to choose. This can be expressed as:

\[ \binom{n}{r} = \frac{n!}{r! (n-r)!} \]
This formula is incredibly useful in solving problems involving selections or counts, such as determining possible triangles.

For instance, if six points are available to choose from, and you need to form triangles, you would apply the combination formula to calculate \( \binom{6}{3} \). This reflects the ways to select sets of three points from the six available, leading to the formation of different triangles.

Exploring Vertex Selection for Triangles

Vertex selection is the process of choosing specific points, or vertices, that will define geometric shapes like triangles. In problems with points, especially in combinatorics, distinct points in the plane play a crucial role in creating various configurations.

In the original exercise example, where six points exist on a plane and no three are collinear, selecting three points at a time will create a single triangle. The identification of different combinations of vertices allows us to ascertain the complete set of potential triangles.

This understanding allows us to explore how multiple triangles can be formed by systematically adjusting which vertices are chosen. It demonstrates how strategic selection is fundamental to solving combinatorial geometry problems.

Collinear Points and Their Impact

Collinear points refer to points that lie on a single straight line. In the context of combinatorial problems, such as forming triangles, it's essential to recognize and treat collinear points differently.

If any set of three points in a given selection is collinear, they cannot form a triangle, as a triangle requires three vertices not on the same line. This distinct characteristic influences how we approach and solve problems involving geometric shapes.

• Collinear points are aligned, limiting their ability to form enclosed shapes like triangles.
• In exercises specifying no three points are collinear, the challenge often involves maximizing valid forms, such as triangles.
• Recognizing whether points are collinear or not helps refine potential solutions in combinatorial contexts.

Understanding the impact of collinear points enables students to navigate problems effectively and ensures accurate combinatorial calculations.