Math Problem Statement

How many triangles can be made by joining 12 points in a plane, given that seven are in one line?

Solution

To find how many triangles can be formed by joining 12 points in a plane, where 7 of these points are collinear (i.e., on the same straight line), we need to break the problem into two steps:

Step 1: Calculate the total number of triangles from 12 points

In general, to form a triangle, we must choose 3 points from the 12 given points. The formula to choose 3 points from nn points is the combination formula: C(n,3)=n!3!(n3)!C(n, 3) = \frac{n!}{3!(n-3)!} For n=12n = 12: [ C(

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Math Problem Analysis

Mathematical Concepts

Combinatorics
Geometry

Formulas

Combination formula: C(n, 3) = n! / [3!(n-3)!]

Theorems

Triangle Formation Theorem

Suitable Grade Level

Grades 9-12