Permutations and CombinationsMCQMTP Sep 24 Series IQuestion 1679 of 249
All Questions

In how many ways can 5 boys and 3 girls sit in a row so that no two girls are together?

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option a — 14,400

All Options:

  • A14,400
  • B14,000
  • C14,425
  • D12,400

Detailed Solution & Explanation

We want to arrange 5 boys and 3 girls in a row such that no two girls sit adjacent to each other. We use the gap method.

1. **Arrange the Boys First:**
We arrange the 5 boys in a row. The number of ways to arrange 5 distinct boys is:
5!=120 ways5! = 120  ways\displaystyle \text{ ways}

2. **Create and Select Gaps for Girls:**
By placing the 5 boys in a row, we create gaps at the ends and between adjacent boys where girls can sit without being together:
_ B1 _ B2 _ B3 _ B4 _ B5 __ \displaystyle \text{\_ } B_1\ _ \displaystyle \text{\_ } B_2\ _ \displaystyle \text{\_ } B_3\ _ \displaystyle \text{\_ } B_4\ _ \displaystyle \text{\_ } B_5\ _\displaystyle \text{\_}
The total number of available gaps is:
5+1=6 gaps.5 + 1 = 6  gaps.\displaystyle \text{ gaps.}
We must select 3 of these 6 gaps and arrange the 3 distinct girls in them. The number of ways to do this is given by permutations:
6P3=6!(6−3)!=6×5×4=120 ways^6P_3 = 6!(6−3)!=6\displaystyle \frac{6!}{(6-3)!} = 6 ×\displaystyle \times 5 ×\displaystyle \times 4 = 120  ways\displaystyle \text{ ways}

3. **Total Arrangements:**
By the fundamental multiplication principle, the total number of seating arrangements is:
Total ways=5!×6P3=120×120=14,400Total ways=5\displaystyle \text{Total ways} = 5! ×\displaystyle \times ^6P_3 = 120 ×\displaystyle \times 120 = 14,400

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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