Permutations and CombinationsMCQMTP Jun 23 - Series IIQuestion 1668 of 251
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The number of 3-digit odd numbers can be formed using the digits 5,6,7,8,9. If repetition is allowed?

Options

A56
B75
C95
D45
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Correct Answer

✅ Option b — 75

All Options:

  • A56
  • B75
  • C95
  • D45

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Detailed Solution & Explanation

We need to form a 3-digit odd number using the digits from the set {5,6,7,8,9}\displaystyle \{5, 6, 7, 8, 9\} when repetition of digits is allowed.

Let the 3-digit number be represented as H‾ T‾ U‾\displaystyle \text{\underline{H}}\ \text{\underline{T}}\ \text{\underline{U}}, where H‾\displaystyle \text{\underline{H}} is the hundreds place, T‾\displaystyle \text{\underline{T}} is the tens place, and U‾\displaystyle \text{\underline{U}} is the units place.

1. **Units Place (U‾\displaystyle \text{\underline{U}}):**
For the number to be odd, the units digit must be an odd number. The odd digits available in our set are {5,7,9}\displaystyle \{5, 7, 9\} (3 options).
Number of ways to fill the units place=3\text{Number of ways to fill the units place} = 3

2. **Tens Place (T‾\displaystyle \text{\underline{T}}):**
Since repetition of digits is allowed, the tens place can be filled by any of the 5 digits {5,6,7,8,9}\displaystyle \{5, 6, 7, 8, 9\} (5 options).
Number of ways to fill the tens place=5\text{Number of ways to fill the tens place} = 5

3. **Hundreds Place (H‾\displaystyle \text{\underline{H}}):**
Similarly, since repetition of digits is allowed, the hundreds place can be filled by any of the 5 digits {5,6,7,8,9}\displaystyle \{5, 6, 7, 8, 9\} (5 options).
Number of ways to fill the hundreds place=5\text{Number of ways to fill the hundreds place} = 5

Using the fundamental multiplication principle, the total number of 3-digit odd numbers that can be formed is:
Total odd numbers=5×5×3=75\text{Total odd numbers} = 5 \times 5 \times 3 = 75

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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