Permutations and CombinationsMCQMTP Dec 22 - Series IQuestion 1730 of 251
All Questions

The number of triangles that can be formed by choosing the vertices from a set of 12 points, Seven of which lie on the same 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

To form a triangle, we need to choose 3\displaystyle 3 non-collinear points from the set of 12\displaystyle 12 points.

If there were no collinear points, the number of ways to choose 3\displaystyle 3 points out of 12\displaystyle 12 would be:
12C3=12×11×103×2×1=220^{12}C_{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220
However, 7\displaystyle 7 of these points lie on the same straight line. Selecting any 3\displaystyle 3 points from these collinear points results in a line segment rather than a triangle. The number of such invalid combinations is:
7C3=7×6×53×2×1=35^{7}C_{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35
Therefore, the number of valid triangles that can be formed is:
Number of Triangles=12C3−7C3\text{Number of Triangles} = ^{12}C_{3} - ^{7}C_{3}
Number of Triangles=220−35=185\text{Number of Triangles} = 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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