Permutations and CombinationsMCQMTP May 18Question 1717 of 251
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15C3r=15Cr+3\displaystyle ^{15}C_{3r} = ^{15}C_{r+3}, then r\displaystyle r is equal to

Options

A2
B3
C4
D5
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Correct Answer

✅ Option b — 3

All Options:

  • A2
  • B3
  • C4
  • D5

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Detailed Solution & Explanation

Given equation:
15C3r=15Cr+3^{15}C_{3r} = ^{15}C_{r+3}
We know that if nCx=nCy\displaystyle ^{n}C_x = ^{n}C_y, then either x=y\displaystyle x = y or x+y=n\displaystyle x + y = n.

Let us check both cases:

**Case 1:** 3r=r+3  ⟹  2r=3  ⟹  r=1.5\displaystyle 3r = r + 3 \implies 2r = 3 \implies r = 1.5 (which is invalid as r\displaystyle r must be an integer).

**Case 2:** 3r+(r+3)=15  ⟹  4r+3=15  ⟹  4r=12  ⟹  r=3\displaystyle 3r + (r + 3) = 15 \implies 4r + 3 = 15 \implies 4r = 12 \implies r = 3.
Thus, mathematically, r\displaystyle r must be 3\displaystyle 3.

**Discrepancy Note:**
The mathematical solution shows that r=3\displaystyle r = 3, which corresponds to **Option B**. The textbook answer key incorrectly lists **Option C** (which is 4\displaystyle 4) as the correct answer. This is a typographical error in the textbook key, as r=4\displaystyle r = 4 would give 15C12=15C7\displaystyle ^{15}C_{12} = ^{15}C_{7}, which is false since 15C12=15C3=455\displaystyle ^{15}C_{12} = ^{15}C_{3} = 455 and 15C7=6435\displaystyle ^{15}C_{7} = 6435.

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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