Permutations and CombinationsMCQPYQ May 18Question 1696 of 251
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The number of triangle that can be formed by choosing the vertices from a set of 12 points, seven of which lie on the same straight line, is:

Options

A185
B175
C115
D105
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Correct Answer

✅ Option a — 185

All Options:

  • A185
  • B175
  • C115
  • D105

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Detailed Solution & Explanation

We need to find the number of triangles that can be formed by choosing 3 vertices from a set of 12 points, where 7 of the points lie on the same straight line (collinear).
1. **Total ways** to choose 3 points out of 12 without any collinearity restriction is:
12C3=12×11×103×2×1=220 ways^{12}C_3 = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \text{ ways}
2. **Unwanted choices**:
Since 7 points lie on the same straight line, any choice of 3 points from these 7 collinear points will lie on a straight line and will not form a triangle.
The number of ways to choose 3 points from these 7 collinear points is:
7C3=7×6×53×2×1=35 ways^7C_3 = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \text{ ways}
3. **Number of triangles formed**:
Number of Triangles=12C3−7C3=220−35=185\text{Number of Triangles} = ^{12}C_3 - ^7C_3 = 220 - 35 = 185
Hence, **Option A** 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.

Key Concepts to Understand

Set

A well-defined collection of distinct objects (called elements or members). Sets are denoted by capital letters and elements are listed within curly braces. Key operations include Union (∪), Intersection (∩), and Complement.

Related Comparison Tables

Permutations vs Combinations: The Quantitative Aptitude Guide

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