Permutations and CombinationsMCQMTP Nov 21Question 1654 of 251
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The number of words that can be formed out of the letters of the word "ARTICLE" so that vowels occupy even places is

Options

A36
B144
C574
D754
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Correct Answer

✅ Option b — 144

All Options:

  • A36
  • B144
  • C574
  • D754

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Detailed Solution & Explanation

The word is ARTICLE, which contains 7 distinct letters: {A,R,T,I,C,L,E}\displaystyle \{A, R, T, I, C, L, E\}.
Let's separate them into vowels and consonants:
- Vowels: {A,I,E}\displaystyle \{A, I, E\} (3 vowels)
- Consonants: {R,T,C,L}\displaystyle \{R, T, C, L\} (4 consonants)
The positions are {1,2,3,4,5,6,7}\displaystyle \{1, 2, 3, 4, 5, 6, 7\}, where:
- Odd places: {1,3,5,7}\displaystyle \{1, 3, 5, 7\} (4 places)
- Even places: {2,4,6}\displaystyle \{2, 4, 6\} (3 places)
We want the vowels to occupy only even places.
Since there are exactly 3 vowels and exactly 3 even places, all 3 vowels must occupy the 3 even positions.
1. Arrange the 3 vowels in the 3 even positions:
Ways=3!=6 ways\text{Ways} = 3! = 6 \text{ ways}
2. Arrange the 4 consonants in the remaining 4 odd positions:
Ways=4!=24 ways\text{Ways} = 4! = 24 \text{ ways}
Total number of arrangements is:
Total=3!×4!=6×24=144\text{Total} = 3! \times 4! = 6 \times 24 = 144
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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