There are 20 points in a plane, how many triangles can be formed by these points if 5 are colinear?

There are 20 points in a plane, how many triangles can be formed by these points if 5 are colinear?

The Correct answer is 1130

Explanation

1. Number of points to choose from for a triangle: We need 3 points to form a triangle.

2. Total number of triangles without considering colinearity: If all 20 points were usable, we could choose 3 points from them in:

20C3 = 20! / (3! * (20-3)!) = 1140

1. Number of triangles formed using colinear points: Since 5 points are colinear, any combination of these 5 points cannot form a triangle.

2. Number of usable points: Therefore, the actual number of points we can choose from to form triangles is 20 - 5 = 15.

3. Number of triangles with usable points: We can now calculate the number of triangles using these 15 points:

15C3 = 15! / (3! * (15-3)!) = 455

1. Total number of triangles formed: Finally, subtract the number of triangles formed using the colinear points from the total number considering all points:

Total triangles = Triangles without colinearity - Triangles with colinearity
= 1140 - 455
= 685

However, the question asks for the number of triangles when non-colinear points are considered. Since using colinear points wouldn't form triangles, we can directly calculate the number of triangles using the following formula:

Number of triangles = 20C3 - 5C3
= 1140 - 10
= 1130

Therefore, there are 1130 triangles that can be formed by joining the non-colinear points.

Colinear in Mathematics

Collinear points are points that lie on the same straight line. If three or more points are collinear, it means that they can all be connected by a single straight line. The term "collinear" comes from the Latin words "con," meaning "together," and "linearis," meaning "pertaining to a line."

In geometry, collinear points play an important role in defining lines, segments, and rays. Three non-collinear points determine a unique plane, while three collinear points do not determine a unique line (since infinitely many lines can pass through three collinear points).

Collinearity can be determined by various methods, such as calculating slopes or using vector equations. In coordinate geometry, if the coordinates of three points A, B, and C are (x₁, y₁), (x₂, y₂), and (x₃, y₃) respectively, then these points are collinear if the slope of the line passing through A and B is equal to the slope of the line passing through A and C, i.e., (y₂ - y₁) / (x₂ - x₁) = (y₃ - y₁) / (x₃ - x₁).