Permutations and CombinationsMCQPYQ Dec 22Question 1699 of 249
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If nPr=3024\displaystyle ^nP_r = 3024 and nCr=126\displaystyle ^nC_r = 126, then find n\displaystyle n and r\displaystyle r?

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Correct Answer

✅ Option a — 9, 4

All Options:

  • A9, 4
  • B10, 3
  • C12, 4
  • D11, 4

Detailed Solution & Explanation

Given the permutation and combination values:
nPr=3024andnCr=126^nP_r = 3024 \quad and\displaystyle \text{and} \quad ^nC_r = 126
Recall the relationship between permutations and combinations:
nPr=r!⋅nCr^nP_r = r! ⋅\displaystyle \cdot ^nC_r
Substituting the given values into the formula:
3024=r!⋅1263024 = r! ⋅\displaystyle \cdot 126
r!=3024126=24r! = 3024126=24\displaystyle \frac{3024}{126} = 24
Since $24 = 4 ×\displaystyle \times 3 ×\displaystyle \times 2 ×\displaystyle \times 1 = 4!,wehave:<br><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>r</mi><mo>=</mo><mn>4</mn></mrow><annotationencoding="application/x−tex">r=4</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</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">4</span></span></span></span></span><br>Now,substitute\displaystyle , we have:<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><mi>r</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">r = 4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</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">4</span></span></span></span></span><br>Now, substituter=4\displaystyle into the permutation formula:<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><mi>n</mi></msup><msub><mi>P</mi><mn>4</mn></msub><mo>=</mo><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo><mo>=</mo><mn>3024</mn></mrow><annotation encoding="application/x-tex">^nP_4 = n(n-1)(n-2)(n-3) = 3024</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8644em;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:0.7144em;"><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 mathnormal mtight">n</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:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal">n</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="mopen">(</span><span class="mord mathnormal">n</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">2</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal">n</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">3</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:0.6444em;"></span><span class="mord">3024</span></span></span></span></span><br>We need to express &#36;3024 as a product of 4 consecutive decreasing integers. Let's test values near 30244≈7.4\displaystyle 30244\displaystyle \sqrt[4]{3024} ≈\displaystyle \approx 7.4:
9×8×7×6=30249 ×\displaystyle \times 8 ×\displaystyle \times 7 ×\displaystyle \times 6 = 3024
This matches perfectly, so we have:
n=9n = 9
Thus, n=9\displaystyle n = 9 and r=4\displaystyle r = 4.
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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