Permutations and CombinationsPYQ Jan 26Question 4563 of 183
All Questions

The total numbers greater than 2000 that can be formed with the digits 1, 2, 3, 4, 5 and no digits being repeated in any number are:

Options

A216
B96
C864
D468
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Correct Answer

✅ Option a — 216

All Options:

  • A216
  • B96
  • C864
  • D468

Detailed Solution & Explanation

We are given the digits: 1,2,3,4,5\displaystyle 1, 2, 3, 4, 5 (total of 5\displaystyle 5 unique digits) with no repetition allowed. We want to form numbers greater than 2000\displaystyle 2000. Such numbers can be either 4\displaystyle 4-digit numbers or 5\displaystyle 5-digit numbers.
**Case 1: 4\displaystyle 4-digit numbers greater than 2000\displaystyle 2000** For a 4\displaystyle 4-digit number to be greater than 2000\displaystyle 2000, the thousands place must be filled by one of the digits: 2,3,4, or 5\displaystyle 2, 3, 4, \text{ or } 5. - Number of choices for the thousands place = 4\displaystyle 4 - The remaining 3\displaystyle 3 positions can be filled using the remaining 4\displaystyle 4 digits without repetition in 4P3\displaystyle ^4P_3 ways: 4P3=4×3×2=24 ways^4P_3 = 4 \times 3 \times 2 = 24\text{ ways} So, the total number of 4\displaystyle 4-digit numbers greater than 2000\displaystyle 2000 is: Number of 4-digit numbers=4×24=96\text{Number of } 4\text{-digit numbers} = 4 \times 24 = 96
**Case 2: 5\displaystyle 5-digit numbers** Any 5\displaystyle 5-digit number formed using these digits will be greater than 2000\displaystyle 2000 (since the smallest 5\displaystyle 5-digit number is 12,345\displaystyle 12,345). The number of ways to arrange all 5\displaystyle 5 digits to form a 5\displaystyle 5-digit number is 5!\displaystyle 5!: 5!=5×4×3×2×1=120 ways5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\text{ ways}
**Total numbers greater than 2000\displaystyle 2000:** Total=96+120=216\text{Total} = 96 + 120 = 216 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.

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