Permutations and CombinationsMCQMTP Jun 23 - Series IQuestion 1667 of 251
All Questions

The number of ways of 4 boys and 3 girls are to be seated for photograph in a row alternatively.

Options

A24
B164
C144
D336
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Correct Answer

✅ Option c — 144

All Options:

  • A24
  • B164
  • C144
  • D336

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Detailed Solution & Explanation

We have 4\displaystyle 4 boys and 3\displaystyle 3 girls to be seated in a row of 7\displaystyle 7 chairs such that they are seated alternatively.

Let us denote the positions in the row as 1,2,3,4,5,6,7\displaystyle 1, 2, 3, 4, 5, 6, 7.
Since the number of boys (4\displaystyle 4) is greater than the number of girls (3\displaystyle 3), the only way to arrange them alternatively is to place the boys in the odd-numbered positions and the girls in the even-numbered positions:
Positions: B‾ G‾ B‾ G‾ B‾ G‾ B‾\text{Positions: } \underline{\text{B}}\ \underline{\text{G}}\ \underline{\text{B}}\ \underline{\text{G}}\ \underline{\text{B}}\ \underline{\text{G}}\ \underline{\text{B}}
Here:
- Boys occupy positions: 1,3,5,7\displaystyle 1, 3, 5, 7 (4 positions).
- Girls occupy positions: 2,4,6\displaystyle 2, 4, 6 (3 positions).

Now, we arrange the individuals:
- The 4 distinct boys can be arranged in the 4 designated boy-positions in 4!\displaystyle 4! ways:
4!=4×3×2×1=24 ways4! = 4 \times 3 \times 2 \times 1 = 24 \text{ ways}
- The 3 distinct girls can be arranged in the 3 designated girl-positions in 3!\displaystyle 3! ways:
3!=3×2×1=6 ways3! = 3 \times 2 \times 1 = 6 \text{ ways}

By the fundamental multiplication principle of counting, the total number of alternate arrangements is:
Total arrangements=4!×3!=24×6=144\text{Total arrangements} = 4! \times 3! = 24 \times 6 = 144

Hence, **Option C** 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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