Permutations and CombinationsMCQPYQ June 19Question 1620 of 251
All Questions

In how many ways can the crew of an eight seated boat be arranged so that 3 of crew can row only on a stroke side and 2 on the other side?

Options

A1,728
B256
C164
D126
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Correct Answer

✅ Option a — 1,728

All Options:

  • A1,728
  • B256
  • C164
  • D126

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Detailed Solution & Explanation

Let us solve this problem step-by-step using combinations and permutations:
An eight-seated boat has two sides: Stroke side (4 seats) and Bow side (4 seats).
Total crew = 8.
- Crew who row only on Stroke side = 3.
- Crew who row only on Bow side = 2.
- Flexible crew (can row on either side) = 8−(3+2)=3\displaystyle 8 - (3 + 2) = 3.

1. **Filling the Stroke side (4 seats):** We already have 3 designated crew members for the Stroke side. We need to choose 1 more crew member from the 3 flexible crew members. This can be done in:
3C1=3textways^3C_1 = 3 \\text{ ways}
2. **Filling the Bow side (4 seats):** The remaining 2 flexible crew members and the 2 designated Bow side rowers will automatically fill the 4 seats on the Bow side (1 way).
3. **Arranging the rowers on each side:**
- The 4 rowers on the Stroke side can be arranged in their 4 seats in 4!=24\displaystyle 4! = 24 ways.
- The 4 rowers on the Bow side can be arranged in their 4 seats in 4!=24\displaystyle 4! = 24 ways.
4. **Total arrangements:** By the multiplication principle, the total number of arrangements is:
textTotalways=3C1times4!times4!=3times24times24=1728\\text{Total ways} = ^3C_1 \\times 4! \\times 4! = 3 \\times 24 \\times 24 = 1728
This mathematically correct answer (1728) corresponds to Option A. However, the textbook key incorrectly lists Option C (164) as correct, which is a typographical error. We proceed with the correct mathematical derivation.
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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