Problem 125 - A Complete Resource Book in Mathematics for JEE Main 2017

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 (D) 120 triangles.

Step 1: Calculate Total Possible Lines

To find the total number of straight lines formed by joining 10 points with no three of them being collinear, use the combination formula to find the number of ways to choose 2 points from 10. This is given by \( \binom{10}{2} = \frac{10 \times 9}{2} = 45 \).

Step 2: Adjust for Collinear Points

Since 4 of the points are collinear, they form only 1 straight line among them, instead of \( \binom{4}{2} = 6 \), so we need to subtract 5 straight lines. This results in \( 45 - 5 = 40 \) lines.

Step 3: Calculate Total Possible Triangles

To find the number of triangles, use the combination formula to find the number of ways to choose 3 points from 10: \( \binom{10}{3} = \frac{10 \times 9 \times 8}{6} = 120 \).

Step 4: Adjust for Collinear Points in Triangles

Triangles cannot be formed by collinear points. There is 1 combination of choosing 3 points from 4 collinear points, which forms a straight line, not a triangle. Thus, subtract 1 from the total: \( 120 - 1 = 119 \).

Key Concepts

Collinear Points

In the study of geometry, especially within the context of combinatorics, it is crucial to understand the concept of collinear points. Collinear points are those that lie on a single straight line. If you think of the points as locations on a map, then collinear points would all fall on the same street.
When dealing with problems involving multiple points, it's important to note that collinear points impact geometric formations, such as lines and triangles. For example, if you have four collinear points, instead of forming six lines (as each pair creates a line), they actually form only one line. Additionally, these four points cannot form any triangles, as a triangle requires three non-collinear points. Thus, understanding which points are collinear helps in accurately calculating geometric outcomes.

Combination Formula

The combination formula is a core tool used in solving combinatorial geometry problems. It helps determine how many ways you can choose a subset of items from a larger set, without considering the order of selection. This formula is expressed generally as \( \binom{n}{r} \), which represents the number of combinations possible when selecting \( r \) items from a total of \( n \) items.
For clarity:

• \( n \) represents the total number of items (or points, in geometry).
• \( r \) represents the subset of items you wish to select.

The mathematical expression for this is \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). It's particularly useful when you need to calculate the number of lines (choosing 2 points) or triangles (choosing 3 points) that can be formed from a set of points. Mastering this formula allows you to solve various problems related to forming lines and triangles efficiently.

Triangle Formation

Forming triangles from a set of points involves understanding which points can be combined. Essentially, a triangle is formed by selecting any three non-collinear points. If three points are collinear, they will form a straight line and not a triangle.
To calculate the possible triangles from a number of points, we apply the combination formula \( \binom{n}{3} \), where \( n \) is the total number of points.
However, when some points are collinear, adjustments must be made. For instance, if there are 4 collinear points, their combinations for triangle formation need to be subtracted from the total since those combinations form lines instead. This approach ensures that all counted triangles are valid and not formed along a straight line. This subtle adjustment highlights the elegance and precision required in combinatorial geometry.