Permutations and CombinationsPYQ Jan 26Question 4262 of 183
All Questions

In a meeting, 5 analysts, 2 consultants, and 3 managers are to be seated in a row. If members of the same profession must sit together, in how many ways can they be seated?

Options

A11,232
B8,640
C6,912
D9,504
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Correct Answer

✅ Option b — 8,640

All Options:

  • A11,232
  • B8,640
  • C6,912
  • D9,504

Detailed Solution & Explanation

Since members of the same profession must sit together, we can treat each profession group as a single entity: 1. One group of 5\displaystyle 5 Analysts 2. One group of 2\displaystyle 2 Consultants 3. One group of 3\displaystyle 3 Managers
These 3\displaystyle 3 groups can be arranged among themselves in 3!\displaystyle 3! ways: 3!=3×2×1=6 ways3! = 3 \times 2 \times 1 = 6\text{ ways}
Within each group, the individuals can be arranged as follows: - The 5\displaystyle 5 analysts can be arranged in 5!\displaystyle 5! ways: 5!=5×4×3×2×1=120 ways5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\text{ ways} - The 2\displaystyle 2 consultants can be arranged in 2!\displaystyle 2! ways: 2!=2×1=2 ways2! = 2 \times 1 = 2\text{ ways} - The 3\displaystyle 3 managers can be arranged in 3!\displaystyle 3! ways: 3!=3×2×1=6 ways3! = 3 \times 2 \times 1 = 6\text{ ways}
Using the multiplication principle, the total number of ways the members can be seated is: Total ways=3!×5!×2!×3!\text{Total ways} = 3! \times 5! \times 2! \times 3! Total ways=6×120×2×6\text{Total ways} = 6 \times 120 \times 2 \times 6 Total ways=720×12=8,640\text{Total ways} = 720 \times 12 = 8,640 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.

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