Permutations and CombinationsMCQMTP Dec 22 Series IIQuestion 1645 of 249
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The Sum of all the 4 digits numbers that can be formed with the digits 3,4,5,5 is

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Correct Answer

✅ Option c — 38887

All Options:

  • A18887
  • B33333
  • C38887
  • D56661

Detailed Solution & Explanation

We need to find the sum of all 4-digit numbers that can be formed using the digits $3, 4, 5, 5.<br>Theavailabledigitsare\displaystyle .<br>The available digits are\{3, 4, 5, 5\}$, where the digit $5isrepeatedtwice.<br>Thetotalnumberofunique4−digitnumbersis:<br><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mtext>Total numbers</mtext><mo>=</mo><mfrac><mrow><mn>4</mn><mostretchy="false">!</mo></mrow><mrow><mn>2</mn><mostretchy="false">!</mo></mrow></mfrac><mo>=</mo><mn>12</mn></mrow><annotationencoding="application/x−tex">Total numbers=4!2!=12</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordtext"><spanclass="mord">Total numbers</span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.0574em;vertical−align:−0.686em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">2</span><spanclass="mclose">!</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">4</span><spanclass="mclose">!</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.686em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">12</span></span></span></span></span><br>Let′sfindthesumofthedigitsateachposition(units,tens,hundreds,thousands):<br>1.∗∗UnitsPosition∗∗:<br>−If\displaystyle is repeated twice.<br>The total number of unique 4-digit numbers is:<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>Total numbers</mtext><mo>=</mo><mfrac><mrow><mn>4</mn><mo stretchy="false">!</mo></mrow><mrow><mn>2</mn><mo stretchy="false">!</mo></mrow></mfrac><mo>=</mo><mn>12</mn></mrow><annotation encoding="application/x-tex">\displaystyle \text{Total numbers} = \frac{4!}{2!} = 12</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">Total 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:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mclose">!</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mclose">!</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></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:0.6444em;"></span><span class="mord">12</span></span></span></span></span><br>Let's find the sum of the digits at each position (units, tens, hundreds, thousands):<br>1. **Units Position**:<br>- If3$ is in the units position, the remaining digits $\{4, 5, 5\}$ can be arranged in $3!2!=3<span\displaystyle \frac{3!}{2!} = 3<span class="katex">ways.So,\displaystyle ways. So,3appears\displaystyle appears3timesintheunitsplace.<br>−If\displaystyle times in the units place.<br>- If4$ is in the units position, the remaining digits $\{3, 5, 5\}$ can be arranged in $3!2!=3<span\displaystyle \frac{3!}{2!} = 3<span class="katex">ways.So,\displaystyle ways. So,4appears\displaystyle appears3timesintheunitsplace.<br>−If\displaystyle times in the units place.<br>- If5$ is in the units position, the remaining digits $\{3, 4, 5\}\displaystyle can be arranged in &#36;3! = 6 ways. So, 5\displaystyle 5 appears 6\displaystyle 6 times in the units place.
Sum of the units digits is:
S=(3×3)+(4×3)+(5×6)=9+12+30=51S = (3 ×\displaystyle \times 3) + (4 ×\displaystyle \times 3) + (5 ×\displaystyle \times 6) = 9 + 12 + 30 = 51
By symmetry, the sum of the digits in the tens, hundreds, and thousands positions is also $51\displaystyle .<br>Therefore, the total sum of all 12 numbers is:<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>Total Sum</mtext><mo>=</mo><mn>51</mn><mo>×</mo><msup><mn>10</mn><mn>3</mn></msup><mo>+</mo><mn>51</mn><mo>×</mo><msup><mn>10</mn><mn>2</mn></msup><mo>+</mo><mn>51</mn><mo>×</mo><msup><mn>10</mn><mn>1</mn></msup><mo>+</mo><mn>51</mn><mo>×</mo><msup><mn>10</mn><mn>0</mn></msup></mrow><annotation encoding="application/x-tex">Total Sum=51\displaystyle \text{Total Sum} = 51 ×\displaystyle \times 10^3 + 51 ×\displaystyle \times 10^2 + 51 ×\displaystyle \times 10^1 + 51 ×\displaystyle \times 10^0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">Total Sum</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">51</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:0.9474em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</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">3</span></span></span></span></span></span></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:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">51</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:0.9474em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</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">2</span></span></span></span></span></span></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:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">51</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:0.9474em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</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">1</span></span></span></span></span></span></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:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">51</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:0.8641em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</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">0</span></span></span></span></span></span></span></span></span></span></span></span><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>Total Sum</mtext><mo>=</mo><mn>51</mn><mo>×</mo><mo stretchy="false">(</mo><mn>1000</mn><mo>+</mo><mn>100</mn><mo>+</mo><mn>10</mn><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>51</mn><mo>×</mo><mn>1111</mn><mo>=</mo><mn>56661</mn></mrow><annotation encoding="application/x-tex">Total Sum=51\displaystyle \text{Total Sum} = 51 ×\displaystyle \times (1000 + 100 + 10 + 1) = 51 ×\displaystyle \times 1111 = 56661</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">Total Sum</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">51</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:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1000</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">100</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">10</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:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</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">51</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">1111</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">56661</span></span></span></span></span><br>Mathematically, the correct sum is &#36;56661 (Option D). However, the textbook answer key contains a typographical error and lists Option C (38887) as correct.
Hence, **Option C** 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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