Permutations and CombinationsMCQMTP June 22Question 1659 of 251
All Questions

In how many ways can the letters of the word "DIRECTOR" be arranged so that the three vowels are never together?

Options

A180
B18,000
C18,002
DNone of these
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Correct Answer

✅ Option b — 18,000

All Options:

  • A180
  • B18,000
  • C18,002
  • DNone of these

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Detailed Solution & Explanation

The word is DIRECTOR, which contains 8 letters in total.
Let's count the frequency of each letter:
- D: 1 time
- I: 1 time (vowel)
- R: 2 times (consonant)
- E: 1 time (vowel)
- C: 1 time (consonant)
- T: 1 time (consonant)
- O: 1 time (vowel)
- R: (already counted)
Vowels: {I,E,O}\displaystyle \{I, E, O\} (3 distinct vowels).
Consonants: {D,R,C,T,R}\displaystyle \{D, R, C, T, R\} (5 consonants, with R\displaystyle R repeating twice).

1. **Total arrangements** of the 8 letters:
Total=8!2!=403202=20160 ways\text{Total} = \frac{8!}{2!} = \frac{40320}{2} = 20160 \text{ ways}
2. **Arrangements where the three vowels are always together**:
Treat the three vowels {I,E,O}\displaystyle \{I, E, O\} as a single block/entity (IEO)\displaystyle (IEO).
Then we have 6 entities: (IEO)\displaystyle (IEO), D\displaystyle D, R\displaystyle R, C\displaystyle C, T\displaystyle T, R\displaystyle R.
Number of ways to arrange these 6 entities (with R\displaystyle R repeated twice) is:
Ways=6!2!=7202=360 ways\text{Ways} = \frac{6!}{2!} = \frac{720}{2} = 360 \text{ ways}
Within the block, the 3 distinct vowels can be arranged in 3!=6\displaystyle 3! = 6 ways.
So, the number of arrangements with vowels together is:
Vowels together=360×6=2160 ways\text{Vowels together} = 360 \times 6 = 2160 \text{ ways}
3. **Arrangements where the three vowels are never together**:
Vowels never together=Total−Vowels together=20160−2160=18000\text{Vowels never together} = \text{Total} - \text{Vowels together} = 20160 - 2160 = 18000
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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