Permutations and CombinationsMCQMTP Dec 22 Series II, PYQ Nov 19Question 1647 of 249
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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?

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option b — 400

All Options:

  • A1200
  • B400
  • C600
  • D1400

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=720Total=6P5=6\displaystyle \text{Total} = ^6P_5 = 6 ×\displaystyle \times 5 ×\displaystyle \times 4 ×\displaystyle \times 3 ×\displaystyle \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).<br>−Theunitspositionisfixedas\displaystyle ).<br>- The units position is fixed as5(1option).<br>−Theremaining4positionscanbefilledusingtheremaining5digits\displaystyle (1 option).<br>- The remaining 4 positions can be filled using the remaining 5 digits\{1, 2, 3, 4, 6\}in\displaystyle in^5P_4\displaystyle ways:<br><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mrow></mrow><mn>5</mn></msup><msub><mi>P</mi><mn>4</mn></msub><mo>=</mo><mn>5</mn><mo>×</mo><mn>4</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>2</mn><mo>=</mo><mn>120</mn><mtext> ways</mtext></mrow><annotation encoding="application/x-tex">^5P_4 = 5 ×\displaystyle \times 4 ×\displaystyle \times 3 ×\displaystyle \times 2 = 120  ways\displaystyle \text{ ways}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0141em;vertical-align:-0.15em;"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">120</span><span class="mord text"><span class="mord"> ways</span></span></span></span></span></span><br>3. **Numbers NOT divisible by 5**:<br><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>Required Numbers</mtext><mo>=</mo><mtext>Total Numbers</mtext><mo>−</mo><mtext>Divisible by 5</mtext><mo>=</mo><mn>720</mn><mo>−</mo><mn>120</mn><mo>=</mo><mn>600</mn></mrow><annotation encoding="application/x-tex">\displaystyle \text{Required Numbers} = \text{Total Numbers} - Divisible by 5=720−120=600\displaystyle \text{Divisible by 5} = 720 - 120 = 600</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">Required Numbers</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord text"><span class="mord">Total Numbers</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">Divisible by 5</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">720</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">120</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">600</span></span></span></span></span><br>Mathematically, the correct answer is &#36;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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