Permutations and CombinationsMCQMTP June 2023 Series IQuestion 1736 of 251
All Questions

The value n,r\displaystyle n, r if nPr=3024\displaystyle ^nP_r = 3024 and nCr=126\displaystyle ^nC_r = 126

Options

A9, 4
B10, 7
C12, 5
D11, 6
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Correct Answer

✅ Option a — 9, 4

All Options:

  • A9, 4
  • B10, 7
  • C12, 5
  • D11, 6

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Detailed Solution & Explanation

We are given:
nPr=3024^nP_r = 3024
nCr=126^nC_r = 126

We know the fundamental relationship between permutations and combinations:
nPr=r!×nCr^nP_r = r! \times ^nC_r

Substituting the given values into this relation:
3024=r!×1263024 = r! \times 126
  ⟹  r!=3024126\implies r! = \frac{3024}{126}
Let us perform the division:
r!=24r! = 24
Since 24=4×3×2×1=4!\displaystyle 24 = 4 \times 3 \times 2 \times 1 = 4!, we find:
r=4r = 4

Now we substitute r=4\displaystyle r = 4 into the permutation formula:
nP4=3024^nP_4 = 3024
  ⟹  n(n−1)(n−2)(n−3)=3024\implies n(n-1)(n-2)(n-3) = 3024
We need to find 4\displaystyle 4 consecutive positive integers whose product is 3024\displaystyle 3024. Let us try n=9\displaystyle n = 9:
9×8×7×6=72×42=30249 \times 8 \times 7 \times 6 = 72 \times 42 = 3024
This matches perfectly. Therefore, n=9\displaystyle n = 9.

Thus, the values are 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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