Permutations and CombinationsMCQPYQ June 24Question 1632 of 251
All Questions

In how many ways the letters of the word "STADIUM" be arranged in such a way that the vowels all occur together?

Options

A7!\displaystyle 7!
B5!4!\displaystyle 5!4!
C5!3!\displaystyle 5!3!
D7!3!\displaystyle 7!3!
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Correct Answer

✅ Option c — 5!3!\displaystyle 5!3!

All Options:

  • A7!\displaystyle 7!
  • B5!4!\displaystyle 5!4!
  • C5!3!\displaystyle 5!3!
  • D7!3!\displaystyle 7!3!

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Detailed Solution & Explanation

Let us analyze the letters of the word **STADIUM**:
- Letters: S, T, A, D, I, U, M (7 distinct letters).
- Vowels = A, I, U (3 vowels).
- Consonants = S, T, D, M (4 consonants).

Since the vowels must always occur together:
1. **Treat the vowels as a single block:** (AIU). We now have 5 items to arrange: the block (AIU) and the 4 consonants.
2. **Arrange the 5 items:** The number of ways is 5!\displaystyle 5!.
3. **Arrange the vowels within the block:** The 3 vowels within the block (AIU) can be arranged in 3!\displaystyle 3! ways.
4. **Total arrangements:** By the multiplication principle, the total number of ways is:
textTotalways=5!times3!\\text{Total ways} = 5! \\times 3!
This matches Option C.
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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