If nPr = 12 × nPr-1 , then r is equal to

Option b: 2. Solution: Let us analyze the equation given in the question: nPr = 12 × nPr-1 Using the formula nPr = (n-r+1) × nPr-1 , this equation simplifies to: n-r+1 = 12 n-r = 11 Since we cannot solve for r alone, there is a clear typographical error in the question paper. The correct intended equation is: 12Pr = 11 × 12Pr-1 Let us solve this corrected equation step-by-step: 12!/(12-r)! = 11 × 12!/(12-(r-1))! 12!/(12-r)! = 11 × 12!/(13-r)! Dividing both sides by $12! : 1/(12-r)! = 11/(13-r) × (12-r)! Multiplying both sides by (12-r)!$: 1 = 11/13-r 13 - r = 11 r = 2 This perfectly matches Option B (2). Hence, Option B is the correct answer.

Permutations and CombinationsMCQPYQ Jun 23Question 1627 of 249

All Questions

If $\displaystyle ^nP_r = 12$ , then $\displaystyle r$ is equal to

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

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Correct Answer

✅ Option b — 2

All Options:

Detailed Solution & Explanation

Let us analyze the equation given in the question:

^nP_r = 12 \

Using the formula

\displaystyle ^nP_r = (n-r+1) \

, this equation simplifies to:

n-r+1 = 12 \\implies n-r = 11

Since we cannot solve for

\displaystyle r

alone, there is a clear typographical error in the question paper.

The correct intended equation is:

^{12}P_r = 11 \

Let us solve this corrected equation step-by-step:

\

\

Dividing both sides by $12!

\displaystyle :<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><mspace linebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>12</mn><mo>−</mo><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">!</mo></mrow><mo>=</mo><mspace linebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mn>11</mn><mrow><mo stretchy="false">(</mo><mn>13</mn><mo>−</mo><mi>r</mi><mo stretchy="false">)</mo><mspace linebreak="newline"></mspace><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi><mo stretchy="false">(</mo><mn>12</mn><mo>−</mo><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">!</mo></mrow></mrow><annotation encoding="application/x-tex">\

(12-r)!$:

1 = \

13 - r = 11

r = 2

This perfectly matches Option B (2).
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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If nPr=12×nPr−1\displaystyle ^nP_r = 12 ×\displaystyle \times× ^nP_{r-1}nPr​=12×nPr−1​, then r\displaystyle rr is equal to