Problem 125 - A Complete Resource Book in Mathematics for JEE Main

Question

There are 10 points in a plane, no three of which are in the same straight line excepting 4 , which are collinear. Then, number of (A) straight lines formed by joining them is 40 (B) triangles formed by joining them is 116 (C) straight lines formed by joining them is 45 (D) triangles formed by joining them is 120

Step-by-Step Solution

Answer

(A) 40 lines and (B) 116 triangles.

Step 1: Calculate Total Original Lines

To find the total number of straight lines that can be formed by 10 points where no three points are collinear, use the combination formula for selecting 2 points out of 10: \( \binom{10}{2} \). This gives: \( \binom{10}{2} = \frac{10 \cdot 9}{2} = 45 \).

Step 2: Adjust for Collinearity

Since 4 points are collinear, they form only 1 line, not 6 lines as \( \binom{4}{2} = 6 \) might suggest. We thus need to subtract these 5 degenerate lines and add back one valid line. Adjusting for this: \( 45 - 5 + 1 = 41 \) lines.

Step 3: Calculate Total Original Triangles

To find the total number of triangles that can be formed by selecting 3 points out of 10, use the combination formula \( \binom{10}{3} \). This gives: \( \binom{10}{3} = \frac{10 \cdot 9 \cdot 8}{3 \cdot 2 \cdot 1} = 120 \).

Step 4: Adjust for Degenerate Triangles

Since the 4 collinear points do not form any triangles, the triangles counted that are degenerate (existing among these 4 points) are \( \binom{4}{3} = 4 \). Subtract these: \( 120 - 4 = 116 \) triangles.

Key Concepts

Collinearity

Collinearity is a geometrical property referring to a set of points lying on a single straight line. In simpler terms, if multiple points sit perfectly along the same line, they are considered collinear. This concept is significant in problems involving points and lines, especially in exercises like the one we're discussing. Consider 10 points on a plane, where typically no three points are collinear. This means most of the points will form multiple lines and triangles without being trapped on a straight path. However, in this exercise, 4 of these 10 points are collinear. As a result, these 4 points only form one single line instead of multiple, showing the importance of recognizing collinearity. Understanding how this affects combination calculations is crucial. When points are collinear, not all potential connections (or lines and triangles they might form) are valid. You subtract the extra, resulting in lines and potential triangles from groups because they don't contribute genuine geometric figures.

Combination Formula

The combination formula is a vital tool in combinatorics, used to determine how many ways you can select a group of items from a larger set without considering the order. It is expressed as \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. In solving the problem, we use this formula frequently:

• To calculate the total number of lines, we find how to choose 2 points from 10, using \( \binom{10}{2} = 45 \).
• To account for lines incorrectly counted due to collinearity, adjustments are made.
• For triangles, picking 3 points out of 10 is essential. This uses \( \binom{10}{3} = 120 \). Again, adjustments are applied for the degenerate triangles caused by collinearity.

Recognizing where and how to apply the combination formula allows for accurate calculation even when collinearity adds complexity.

Triangle Formation

Understanding triangle formation involves recognizing how three non-collinear points form a triangle. Each set of points that aren't on the same line can serve as a vertex, thereby contributing to a triangular shape.In our exercise, we start with 10 points. If none are collinear, we calculate potential triangles using the combination formula: \( \binom{10}{3} = 120 \). This number assumes every group of three points forms a triangle, but that's not the whole story.Since 4 of these points are collinear, they don't create any real triangles if used together. In fact, the supposed triangles involving these 4 points are invalid or degenerate. So, we subtract the sets that don't genuinely form triangles, i.e., \( \binom{4}{3} = 4 \) degenerate triangles, leading us to the correct count of 116 valid triangles. Thus, thorough understanding of which points can form triangles helps ensure precise results, especially when collinearity comes into play.