Permutations and CombinationsMCQMTP Dec 22 Series IIQuestion 1716 of 251
All Questions

In how many ways can 4 people be selected at random from 6 boys and 4 girls if there are exactly two girls?

Options

A90
B360
C92
D480
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Correct Answer

✅ Option a — 90

All Options:

  • A90
  • B360
  • C92
  • D480

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Detailed Solution & Explanation

We are required to select a group of 4\displaystyle 4 people from a pool of 6\displaystyle 6 boys and 4\displaystyle 4 girls such that the group contains **exactly two girls**.

Since the group consists of 4\displaystyle 4 people in total, and exactly 2\displaystyle 2 must be girls, the remaining 4−2=2\displaystyle 4 - 2 = 2 members of the group must be boys.

Thus, the task consists of two independent selections:
1. Selecting 2\displaystyle 2 girls out of 4\displaystyle 4 available girls.
2. Selecting 2\displaystyle 2 boys out of 6\displaystyle 6 available boys.

Let us calculate the number of ways for each selection:
- The number of ways to choose 2\displaystyle 2 girls from 4\displaystyle 4 is given by 4C2\displaystyle ^{4}C_{2}:
4C2=4!2!(4−2)!=4×32×1=6^{4}C_{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6
- The number of ways to choose 2\displaystyle 2 boys from 6\displaystyle 6 is given by 6C2\displaystyle ^{6}C_{2}:
6C2=6!2!(6−2)!=6×52×1=15^{6}C_{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15

By the fundamental multiplication principle of counting, the total number of ways to form the group is:
Total Ways=4C2×6C2=6×15=90\text{Total Ways} = ^{4}C_{2} \times ^{6}C_{2} = 6 \times 15 = 90

**Discrepancy Note:**
The mathematical derivation yields 90\displaystyle 90 ways, which corresponds to **Option A**. The textbook answer key marks **Option C** (92\displaystyle 92) as correct, which is mathematically incorrect and represents a typographical error in the source material.

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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