Permutations and CombinationsMCQPYQ June 24Question 1633 of 251
All Questions

How many ways can 5 different trophies can be arranged on a shelf if one particular trophy must always be at 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

Let us solve this problem step-by-step:
We have 5 different trophies to arrange on a shelf, and one particular trophy must always be placed in the middle (3rd position).

1. **Place the particular trophy:** The middle seat is fixed for this 1 trophy (1 choice).
2. **Arrange the remaining trophies:** The remaining 5−1=4\displaystyle 5 - 1 = 4 trophies can be arranged in the remaining 4 positions in:
4!=24textways4! = 24 \\text{ ways}
3. **Total arrangements:** The total number of ways to arrange the shelf is:
1times24=241 \\times 24 = 24
This matches Option A.
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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