Permutations and CombinationsMCQMTP Dec 22 Series II, PYQ Nov 19Question 1647 of 251
All Questions

How many numbers can be formed with the help of 2, 3, 4, 5, 6, 1 which is not divisible by 5, given that it is a five-digit number and digits are not repeating?

Options

A1200
B400
C600
D1400
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Correct Answer

✅ Option b — 400

All Options:

  • A1200
  • B400
  • C600
  • D1400

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Detailed Solution & Explanation

We want to find how many five-digit numbers can be formed using the digits {2,3,4,5,6,1}\displaystyle \{2, 3, 4, 5, 6, 1\} (6 distinct digits) without repetition such that they are NOT divisible by 5.
1. **Total five-digit numbers** that can be formed from 6 digits without repetition is:
Total=6P5=6×5×4×3×2=720\text{Total} = ^6P_5 = 6 \times 5 \times 4 \times 3 \times 2 = 720
2. **Numbers divisible by 5**:
For a five-digit number to be divisible by 5, its units digit must be 5\displaystyle 5 (since the only available digits are 2,3,4,5,6,1\displaystyle 2, 3, 4, 5, 6, 1).
- The units position is fixed as 5\displaystyle 5 (1 option).
- The remaining 4 positions can be filled using the remaining 5 digits {1,2,3,4,6}\displaystyle \{1, 2, 3, 4, 6\} in 5P4\displaystyle ^5P_4 ways:
5P4=5×4×3×2=120 ways^5P_4 = 5 \times 4 \times 3 \times 2 = 120 \text{ ways}
3. **Numbers NOT divisible by 5**:
Required Numbers=Total Numbers−Divisible by 5=720−120=600\text{Required Numbers} = \text{Total Numbers} - \text{Divisible by 5} = 720 - 120 = 600
Mathematically, the correct answer is 600\displaystyle 600 (Option C). However, the textbook answer key contains a typographical error and lists Option B (400) as correct.
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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