Permutations and CombinationsMCQPYQ July 21Question 1612 of 251
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How many no. of seven-digit numbers which can be formed from the digits 3, 4, 5, 6, 7, 8, 9 no digits being repeated are not divisible by 5?

Options

A4320
B4690
C3900
D3890
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Correct Answer

✅ Option a — 4320

All Options:

  • A4320
  • B4690
  • C3900
  • D3890

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Detailed Solution & Explanation

Let us solve the problem step-by-step using permutations:
The given digits are: {3,4,5,6,7,8,9}\displaystyle \{3, 4, 5, 6, 7, 8, 9\} (total 7 distinct digits). We want to form a 7-digit number without repetition that is **not** divisible by 5.

1. **Total possible 7-digit numbers:** The number of ways to arrange 7 distinct digits is:
textTotal=7!=5040\\text{Total} = 7! = 5040
2. **7-digit numbers divisible by 5:** A number is divisible by 5 if its unit's digit is 5. Fixing 5 in the unit's place (1 choice), the remaining 6 positions can be filled using the remaining 6 digits in:
textDivisibleby5=6!=720\\text{Divisible by 5} = 6! = 720
3. **7-digit numbers not divisible by 5:** The number of 7-digit numbers not divisible by 5 is:
textNotdivisibleby5=textTotal−textDivisibleby5\\text{Not divisible by 5} = \\text{Total} - \\text{Divisible by 5}
textNotdivisibleby5=5040−720=4320\\text{Not divisible by 5} = 5040 - 720 = 4320
This matches 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.

Related Comparison Tables

Permutations vs Combinations: The Quantitative Aptitude Guide

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