Permutations and CombinationsMCQMTP June 24 Series IIQuestion 1742 of 251
All Questions

A Supreme Court Bench consists of 5\displaystyle 5 judges. In how many ways, the bench can give a majority division?

Options

A10\displaystyle 10
B5\displaystyle 5
C15\displaystyle 15
D16\displaystyle 16
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Correct Answer

✅ Option d — 16\displaystyle 16

All Options:

  • A10\displaystyle 10
  • B5\displaystyle 5
  • C15\displaystyle 15
  • D16\displaystyle 16

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Detailed Solution & Explanation

A Supreme Court Bench consists of 5\displaystyle 5 judges. A majority decision (or division) requires that **at least 3 judges** agree.

This means the majority can consist of:
1. Exactly 3\displaystyle 3 judges
2. Exactly 4\displaystyle 4 judges
3. Exactly 5\displaystyle 5 judges

Let us calculate the number of ways to choose these judges:
- Choosing 3\displaystyle 3 judges out of 5\displaystyle 5:
5C3=5×42×1=10 ways^{5}C_{3} = \frac{5 \times 4}{2 \times 1} = 10 \text{ ways}
- Choosing 4\displaystyle 4 judges out of 5\displaystyle 5:
5C4=5 ways^{5}C_{4} = 5 \text{ ways}
- Choosing 5\displaystyle 5 judges out of 5\displaystyle 5:
5C5=1 way^{5}C_{5} = 1 \text{ way}
Summing these up, the total number of ways the bench can give a majority decision is:
Total Ways=5C3+5C4+5C5\text{Total Ways} = ^{5}C_{3} + ^{5}C_{4} + ^{5}C_{5}
Total Ways=10+5+1=16\text{Total Ways} = 10 + 5 + 1 = 16

Hence, **Option D** 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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