nP3 = 2 × nP2 . Find n .

Option a: 4. Solution: Let us write down the equation using the definition of permutations: nP3 = 2 × nP2 Expanding both sides using the permutation formula nPr = n(n-1)...(n-r+1) : n(n-1)(n-2) = 2 × n(n-1) Since n ≥ 3 for nP3 to be defined, n(n-1) ≠ 0 . Therefore, we can divide both sides by n(n-1) : n - 2 = 2 n = 4 This matches Option A. Hence, Option A is the correct answer.

$^nP_3 = 2 \cdot ^nP_2$. Find $n$.

The correct answer is Option a: 4. Let us write down the equation using the definition of permutations: $$^nP_3 = 2 \\cdot ^nP_2$$ Expanding both sides using the permutation formula $^nP_r = n(n-1)\\dots(n-r+1)$: $$n(n-1)(n-2) = 2 \\cdot n(n-1)$$ Since $n \\ge 3$ for $^nP_3$ to be defined, $n(n-1) \\neq 0$. Therefore, we can divide both sides by $n(n-1)$: $$n - 2 = 2$$ $$n = 4$$ This matches Option A. Hence, **Option A** is the correct answer.

Permutations and CombinationsMCQPYQ Nov. 19Question 1604 of 251

All Questions

$\displaystyle ^nP_3 = 2 \cdot ^nP_2$ . Find $\displaystyle n$ .

Options

\displaystyle 7/2

\displaystyle 2/7

For any discrepancies in this question, email contact@cadada.in

Correct Answer

✅ Option a — 4

All Options:

A4
B $\displaystyle 7/2$
C5
D $\displaystyle 2/7$

Detailed Solution & Explanation

Let us write down the equation using the definition of permutations:

^nP_3 = 2 \\cdot ^nP_2

Expanding both sides using the permutation formula

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

n(n-1)(n-2) = 2 \\cdot n(n-1)

Since

\displaystyle n \\ge 3

for

\displaystyle ^nP_3

to be defined,

\displaystyle n(n-1) \\neq 0

. Therefore, we can divide both sides by

\displaystyle n(n-1)

n - 2 = 2

n = 4

This matches Option A.
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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nP3=2⋅nP2\displaystyle ^nP_3 = 2 \cdot ^nP_2nP3​=2⋅nP2​. Find n\displaystyle nn.