Permutations and CombinationsMCQMTP Dec 23 - Series IIQuestion 1674 of 251
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Word 'REGULATION' is arranged without repetition. Find the probability that the vowels come at odd places.

Options

A1
B1
C252
D144
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Correct Answer

✅ Option a — 1

All Options:

  • A1
  • B1
  • C252
  • D144

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Detailed Solution & Explanation

We want to find the probability that the vowels occupy only the odd places when the letters of the word "REGULATION" are arranged randomly without repetition.

First, let us analyze the letters in the word "REGULATION":
- Total number of letters = 10.
- Vowels: {E,U,A,I,O}\displaystyle \{E, U, A, I, O\} (5 vowels).
- Consonants: {R,G,L,T,N}\displaystyle \{R, G, L, T, N\} (5 consonants).

In a 10-letter arrangement, the positions are numbered 1,2,3,…,10\displaystyle 1, 2, 3, \dots, 10.
- Odd places: 1,3,5,7,9\displaystyle 1, 3, 5, 7, 9 (exactly 5 positions).
- Even places: 2,4,6,8,10\displaystyle 2, 4, 6, 8, 10 (exactly 5 positions).

For the vowels to occupy the odd positions:
- The 5 vowels must be arranged in the 5 odd places. The number of ways to do this is 5!=120\displaystyle 5! = 120.
- The 5 consonants must be arranged in the remaining 5 even places. The number of ways to do this is 5!=120\displaystyle 5! = 120.

So, the number of favorable arrangements is:
Favorable arrangements=5!×5!=120×120=14,400\text{Favorable arrangements} = 5! \times 5! = 120 \times 120 = 14,400

The total number of unrestricted arrangements of the 10 distinct letters is:
Total arrangements=10!=3,628,800\text{Total arrangements} = 10! = 3,628,800

The probability P\displaystyle P of this event occurring is:
P=Favorable arrangementsTotal arrangements=5!×5!10!P = \frac{\text{Favorable arrangements}}{\text{Total arrangements}} = \frac{5! \times 5!}{10!}
P=120×1203,628,800=14,4003,628,800=1252P = \frac{120 \times 120}{3,628,800} = \frac{14,400}{3,628,800} = \frac{1}{252}

In the provided options, the value 1/252\displaystyle 1/252 is represented as Option A.

Hence, **Option A** 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.

Key Concepts to Understand

Probability

A numerical measure of the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). Formula: P(A) = Number of favorable outcomes / Total number of equally likely outcomes.

Related Comparison Tables

Permutations vs Combinations: The Quantitative Aptitude Guide

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