Permutations and CombinationsMCQMTP May 18Question 1648 of 251
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If two letters are taken at random from the word HOME, what is the Probability that none of the letters would be vowels?

Options

A16\displaystyle \frac{1}{6}
B12\displaystyle \frac{1}{2}
C13\displaystyle \frac{1}{3}
D14\displaystyle \frac{1}{4}
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Correct Answer

✅ Option a — 16\displaystyle \frac{1}{6}

All Options:

  • A16\displaystyle \frac{1}{6}
  • B12\displaystyle \frac{1}{2}
  • C13\displaystyle \frac{1}{3}
  • D14\displaystyle \frac{1}{4}

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Detailed Solution & Explanation

The word is HOME, which contains 4 letters: {H,O,M,E}\displaystyle \{H, O, M, E\}.
Let's separate the letters into vowels and consonants:
- Vowels: {O,E}\displaystyle \{O, E\} (2 vowels)
- Consonants: {H,M}\displaystyle \{H, M\} (2 consonants)
We select two letters at random from the 4 letters.
The total number of ways to choose 2 letters from 4 is:
Total outcomes=4C2=4×32=6 ways\text{Total outcomes} = ^4C_2 = \frac{4 \times 3}{2} = 6 \text{ ways}
We want the probability that none of the chosen letters is a vowel. This means both chosen letters must be consonants.
The number of ways to choose 2 consonants from the 2 available consonants is:
Favorable outcomes=2C2=1 way\text{Favorable outcomes} = ^2C_2 = 1 \text{ way}
Therefore, the probability is:
Probability=Favorable outcomesTotal outcomes=16\text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{1}{6}
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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