Permutations and CombinationsMCQMTP Apr 21Question 1721 of 249
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The Supreme Court Bench consists of 5 judges. In how many ways, the bench can give a majority decision?

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option d — 16

All Options:

  • A10
  • B5
  • C15
  • D16

Detailed Solution & Explanation

A Supreme Court Bench consists of 5\displaystyle 5 judges. A majority decision requires that **at least 3 judges** agree on the decision.

This means the decision can be supported by:
1. Exactly 3\displaystyle 3 judges
2. Exactly 4\displaystyle 4 judges
3. Exactly 5\displaystyle 5 judges

The number of ways to select the judges who form the majority in favor of the decision is:
- Choosing 3\displaystyle 3 judges out of 5\displaystyle 5:
5C3=5!3!×2!=5×42×1=10^{5}C_{3} = \displaystyle \frac{5!}{3! \times 2!} = \frac{5 ×\displaystyle \times 4}{2 ×\displaystyle \times 1} = 10
- Choosing 4\displaystyle 4 judges out of 5\displaystyle 5:
5C4=5!4!×1!=5^{5}C_{4} = 5!4!×1!=5\displaystyle \frac{5!}{4! \times 1!} = 5
- Choosing 5\displaystyle 5 judges out of 5\displaystyle 5:
5C5=5!5!×0!=1^{5}C_{5} = 5!5!×0!=1\displaystyle \frac{5!}{5! \times 0!} = 1

Summing these up, the total number of ways the bench can give a majority decision (in favor of a particular outcome) is:
Total Ways=5C3+5C4+5C5Total Ways=5C3+5C4+5C5\displaystyle \text{Total Ways} = ^{5}C_{3} + ^{5}C_{4} + ^{5}C_{5}
Total Ways=10+5+1=16Total Ways=10+5+1=16\displaystyle \text{Total Ways} = 10 + 5 + 1 = 16

**Discrepancy Note:**
The mathematically correct number of ways to give a majority decision is $16\displaystyle , which corresponds to **Option D**. The textbook answer key for this specific mock test paper has a typographical error, indicating **Option A** ($10) as the correct answer (which only counts the case of exactly 3 judges agreeing). Other occurrences of this exact question in the textbook correctly identify **Option D** as the correct answer.

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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