Permutations and CombinationsMCQMTP Sep 24 Series IIQuestion 1680 of 251
All Questions

How many ways can 5 different trophies can be arranged on a shelf if one particular trophy must always be in the middle?

Options

A24
B120
C48
D144
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Correct Answer

✅ Option a — 24

All Options:

  • A24
  • B120
  • C48
  • D144

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Detailed Solution & Explanation

We have 5 distinct trophies to be arranged in 5 positions on a shelf.
The condition is that one particular trophy must always occupy the middle position (Position 3).

1. **Middle Position:**
The middle position (Position 3) is reserved for that one particular trophy. There is exactly 1 way to place it there.
Ways to fill the middle position=1\text{Ways to fill the middle position} = 1

2. **Remaining Positions:**
The remaining 5−1=4\displaystyle 5 - 1 = 4 trophies must be arranged in the remaining 4 vacant positions on the shelf. The number of ways to arrange 4 distinct objects in a row is:
4!=4×3×2×1=24 ways4! = 4 \times 3 \times 2 \times 1 = 24 \text{ ways}

By the fundamental multiplication principle of counting, the total number of arrangements is:
Total arrangements=1×24=24\text{Total arrangements} = 1 \times 24 = 24

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.

Related Comparison Tables

Permutations vs Combinations: The Quantitative Aptitude Guide

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