Permutations and CombinationsMCQPYQ Dec 22Question 1698 of 251
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There are 20 points in a plane area. How many triangles can be formed by these points if 5 points are collinear?

Options

A550
B560
C1130
D1140
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Correct Answer

✅ Option c — 1130

All Options:

  • A550
  • B560
  • C1130
  • D1140

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Detailed Solution & Explanation

We have 20 points in a plane, of which 5 points are collinear. We want to find the number of triangles that can be formed.
A triangle is formed by choosing any 3 non-collinear points.
1. **Total ways** to choose 3 points out of 20 without restrictions is:
20C3=20×19×183×2×1=1140 ways^{20}C_3 = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} = 1140 \text{ ways}
2. **Unwanted choices**:
Since 5 points are collinear, choosing 3 points from these 5 collinear points will only result in a straight line, not a triangle.
Number of ways to choose 3 points from the 5 collinear points is:
5C3=5×42=10 ways^5C_3 = \frac{5 \times 4}{2} = 10 \text{ ways}
3. **Number of triangles formed**:
Number of Triangles=20C3−5C3=1140−10=1130\text{Number of Triangles} = ^{20}C_3 - ^5C_3 = 1140 - 10 = 1130
Hence, **Option C** is the correct answer.

About This Chapter: Permutations and Combinations

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Factorials, Permutations, Combinations

This chapter deals with the fundamental principles of counting. It covers factorials, circular permutations, restricted permutations, combinations, and the differences between selecting items versus arranging them.

View Official ICAI Syllabus

Exam Strategy Tip

The most common mistake is confusing 'P' (Arrangement) with 'C' (Selection). If order matters (like opening a lock), use P. If order doesn't matter (like choosing a team), use C.

Related Comparison Tables

Permutations vs Combinations: The Quantitative Aptitude Guide

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