Permutations and CombinationsMCQMTP Dec 23 - Series IIQuestion 1673 of 251
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There are 5 books on English, 4 books on Tamil and 3 books on Hindi. In how many ways can these books be placed on a shelf if the books on the same subjects are to be together?

Options

A1,36,800
B1,03,600
C1,03,680
D1,63,800
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Correct Answer

✅ Option c — 1,03,680

All Options:

  • A1,36,800
  • B1,03,600
  • C1,03,680
  • D1,63,800

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Detailed Solution & Explanation

We have a total of 12 books belonging to three different subjects:
- English (5 books)
- Tamil (4 books)
- Hindi (3 books)

We want to place these books on a shelf such that the books on the same subject are always grouped together.

1. **Arrangement of Subject Blocks:**
Since books on the same subject must be together, we treat each subject group as a single block: Block E\displaystyle E, Block T\displaystyle T, and Block H\displaystyle H.
The number of ways to arrange these 3 blocks on the shelf is:
3!=3×2×1=6 ways3! = 3 \times 2 \times 1 = 6 \text{ ways}

2. **Internal Arrangement of English Books:**
The 5 distinct English books can be arranged among themselves inside Block E\displaystyle E in:
5!=5×4×3×2×1=120 ways5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \text{ ways}

3. **Internal Arrangement of Tamil Books:**
The 4 distinct Tamil books can be arranged among themselves inside Block T\displaystyle T in:
4!=4×3×2×1=24 ways4! = 4 \times 3 \times 2 \times 1 = 24 \text{ ways}

4. **Internal Arrangement of Hindi Books:**
The 3 distinct Hindi books can be arranged among themselves inside Block H\displaystyle H in:
3!=3×2×1=6 ways3! = 3 \times 2 \times 1 = 6 \text{ ways}

Using the fundamental multiplication principle of counting, the total number of arrangements is:
Total ways=3!×5!×4!×3!=6×120×24×6\text{Total ways} = 3! \times 5! \times 4! \times 3! = 6 \times 120 \times 24 \times 6
Total ways=720×144=103,680\text{Total ways} = 720 \times 144 = 103,680

Hence, **Option C** 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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