Permutations and CombinationsMCQPYQ Jan. 21Question 1606 of 251
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Eight chairs are numbered from 1 to 8. Two women and three men are to be seated by allowing one chair for each. First, the women choose the chairs from the chairs numbered 1 to 4 and then men select the chairs from the remaining. The number of possible arrangements is:

Options

A120
B288
C32
D1440
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Correct Answer

✅ Option d — 1440

All Options:

  • A120
  • B288
  • C32
  • D1440

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Detailed Solution & Explanation

Let us solve the problem step-by-step using the multiplication principle of counting:
1. **Seating the women:** There are 2 women and they must choose from chairs numbered 1 to 4 (4 chairs). Since each person is distinct, the number of ways to seat them is:
4P2=frac4!(4−2)!=4times3=12textways^4P_2 = \\frac{4!}{(4-2)!} = 4 \\times 3 = 12 \\text{ ways}
2. **Seating the men:** There are 3 men. The total number of chairs is 8, and 2 chairs have already been occupied by the women, leaving 8−2=6\displaystyle 8 - 2 = 6 chairs. The 3 men can choose and arrange themselves in these 6 remaining chairs in:
6P3=frac6!(6−3)!=6times5times4=120textways^6P_3 = \\frac{6!}{(6-3)!} = 6 \\times 5 \\times 4 = 120 \\text{ ways}
3. **Total arrangements:** By the multiplication principle, the total number of seating arrangements is:
textTotalways=12times120=1440\\text{Total ways} = 12 \\times 120 = 1440
This mathematically correct answer (1440) corresponds to Option D. However, the textbook key incorrectly lists Option A (120) as correct, which represents only the arrangements for the men (6P3=120\displaystyle ^6P_3 = 120), forgetting to multiply by the women's seating arrangements (12).
Hence, **Option D** 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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