Permutations and CombinationsMCQMTP June 2023 Series IQuestion 1735 of 251
All Questions

If there are 30 points in a plane of which 5 points are lies on the same line. Then the number of triangles can be formed?

Options

A650
B580
C4050
D4060
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Correct Answer

✅ Option c — 4050

All Options:

  • A650
  • B580
  • C4050
  • D4060

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Detailed Solution & Explanation

To form a triangle, we need to choose 3\displaystyle 3 non-collinear points from the given set of 30\displaystyle 30 points.

If all 30\displaystyle 30 points were non-collinear, the total number of ways to choose 3\displaystyle 3 points would be:
30C3=30!3!(30−3)!=30×29×283×2×1=10×29×14=4060^{30}C_{3} = \frac{30!}{3!(30-3)!} = \frac{30 \times 29 \times 28}{3 \times 2 \times 1} = 10 \times 29 \times 14 = 4060
However, we are given that 5\displaystyle 5 of these points are collinear (they lie on the same straight line). Selecting any 3\displaystyle 3 points from these 5\displaystyle 5 collinear points will lie on a straight line and cannot form a triangle. The number of such invalid selections is:
5C3=5!3!×2!=5×42×1=10^{5}C_{3} = \frac{5!}{3! \times 2!} = \frac{5 \times 4}{2 \times 1} = 10
Subtracting the invalid combinations from the total combinations gives the number of valid triangles:
Number of Triangles=30C3−5C3\text{Number of Triangles} = ^{30}C_{3} - ^{5}C_{3}
Number of Triangles=4060−10=4050\text{Number of Triangles} = 4060 - 10 = 4050

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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