Question

A plane contains 20 points of which 6 are collinear. How many different triangles can be formed with these points? (1) 1120 (2) 1140 (3) 1121 (4) 1139

Short answer

Answer: 1120 different triangles can be formed using 20 points on a plane, where 6 of them are collinear.

Step-by-step solution

Find the total number of combinations with 20 points

We need to find the total number of combinations that can be formed by choosing 3 points from the given 20 points. This can be found using Combinations formula which is: C(n, k) = n! / (k! * (n - k)!) Here, n = 20 (total points) and k = 3 (points to form a triangle). So, C(20, 3) = 20! / (3! * (20 - 3)!) = 20! / (3! * 17!) = 1140

Find the number of combinations with 6 collinear points

Now we must find the number of combinations that can be formed by choosing 3 points from the given 6 collinear points. This can be found using the same Combinations formula as above: C(6, 3) = 6! / (3! * (6 - 3)!) = 6! / (3! * 3!) = 20

Subtract the number of invalid combinations

Now, to find the number of valid combinations (i.e. triangles) we must subtract the number of combinations obtained from the collinear points from the total combinations: Valid combinations = Total combinations - Invalid combinations = 1140 - 20 = 1120 So, the answer is (1) 1120.

Key concepts

Collinear Points

Collinear points refer to a set of points that lie on the same straight line. In geometry, this concept is crucial as it determines the shape and dimensions that can be formed with a given set of points. For example, when we speak of forming triangles from a set of points, we must consider that any three collinear points cannot form a triangle, because they don't span an area.

Specifically, in the context of the given exercise, we have 6 points that are collinear. The significance of identifying these collinear points becomes apparent when we want to calculate how many triangles can be formed. Triangles, by definition, are a closed three-sided figure with non-collinear vertices; therefore, selecting three collinear points would not satisfy the conditions for a triangle and would result in a line segment instead of a triangular shape. This distinction is essential for finding a correct solution to our combinatoric problem.

Triangle Combinations

To understand triangle combinations, we delve into the specifics of how triangles can be formed with a given set of points. A triangle is formed by connecting three non-collinear points, which means each triangle can be uniquely identified by its three vertices. When calculating the number of triangle combinations possible from a plane containing multiple points, it is vital to consider all possible selections of three points that satisfy the non-collinearity requirement.

Formation of Unique Triangles

When determining combinations for triangles, one must ensure that each combination of three points taken from the set is unique; that is no two combinations should repeat the same set of points. This approach helps us in avoiding the counting of duplicates and in maintaining the integrity of the solution, which is crucial for accurate calculation in combinatorial problems.

Permutations and Combinations

The concepts of permutations and combinations are foundational in combinatorial mathematics, which deals with counting and arranging objects. While permutations are concerned with arrangements of objects in a specific order, combinations are focused on selections where the order does not matter.

Using the combinations formula \(C(n, k) = \frac{n!}{k! \cdot (n - k)!}\) allows us to calculate the number of ways we can select k items from a set of n items without regard to order. This is especially relevant in our exercise, where we need to select 3 points out of 20 to form a triangle, and the order in which these points are selected is immaterial; whether we choose points A, B, and C or C, A, and B first, the resulting triangle remains the same.

Selecting Subsets

In these scenarios, it is not only important to know how to use the combinations formula but also to understand its practical implications. When subtracting invalid combinations, such as those formed by collinear points, we are honing in on the viable subsets of points that can form the triangles we're counting. This distinction between permutations and combinations, and the inherent understanding of when to apply each, is critical for solving a wide range of mathematical problems accurately.