Permutations and CombinationsMCQMTP Nov 21Question 1655 of 251
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A box contains 3 pink caps, 2 purple caps and 4 orange caps. In how many ways the caps can be arranged so that the caps of the same colour are together. (Assume all caps of same colour are not identical)

Options

A1724
B1728
C1732
D1764
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Correct Answer

✅ Option b — 1728

All Options:

  • A1724
  • B1728
  • C1732
  • D1764

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Detailed Solution & Explanation

We have:
- 3 pink caps
- 2 purple caps
- 4 orange caps
Since the caps of the same color are not identical, each of the 3+2+4=9\displaystyle 3 + 2 + 4 = 9 caps is distinct.
We want to arrange them in a row such that caps of the same color are always together.
1. Treat the caps of each color as a single block: one block of pink caps, one block of purple caps, and one block of orange caps. This gives us 3 blocks.
Number of ways to arrange the 3 blocks = 3!=6\displaystyle 3! = 6 ways.
2. Within the pink block, the 3 distinct pink caps can be arranged in 3!=6\displaystyle 3! = 6 ways.
3. Within the purple block, the 2 distinct purple caps can be arranged in 2!=2\displaystyle 2! = 2 ways.
4. Within the orange block, the 4 distinct orange caps can be arranged in 4!=24\displaystyle 4! = 24 ways.
Using the multiplication principle, the total number of arrangements is:
Total=3!×3!×2!×4!=6×6×2×24=1728\text{Total} = 3! \times 3! \times 2! \times 4! = 6 \times 6 \times 2 \times 24 = 1728
Hence, **Option B** 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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