Permutations and CombinationsMCQPYQ Jan. 21Question 1614 of 251
All Questions

In how many different ways can the letters of the word 'DETAIL' be arranged so that the vowels occupy only the odd positions?

Options

A32
B36
C48
D60
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Correct Answer

✅ Option b — 36

All Options:

  • A32
  • B36
  • C48
  • D60

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Detailed Solution & Explanation

Let us analyze the letters of the word **DETAIL**:
- Total letters = 6 (D, E, T, A, I, L).
- Vowels = E, A, I (3 vowels).
- Consonants = D, T, L (3 consonants).
- Total positions = 6 (1st, 2nd, 3rd, 4th, 5th, 6th).
- Odd positions = 1st, 3rd, 5th (3 positions).
- Even positions = 2nd, 4th, 6th (3 positions).

The vowels must occupy only the odd positions:
1. Arrange the 3 vowels in the 3 odd positions in 3!=6\displaystyle 3! = 6 ways.
2. Arrange the 3 consonants in the remaining 3 even positions in 3!=6\displaystyle 3! = 6 ways.
3. By the multiplication principle, the total number of ways is:
textTotalways=3!times3!=6times6=36\\text{Total ways} = 3! \\times 3! = 6 \\times 6 = 36
This mathematically correct answer (36) corresponds to Option B. However, the textbook key incorrectly lists Option A (32) as correct, which is a typographical error. We proceed with the correct mathematical derivation.
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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