Permutations and CombinationsMCQMTP June 24 Series IIQuestion 1678 of 251
All Questions

A room has 10 doors. In how many ways can a man enter the room by one door and come out by a different door.

Options

A90
B100
C50
DNone of these
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Correct Answer

✅ Option a — 90

All Options:

  • A90
  • B100
  • C50
  • DNone of these

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Detailed Solution & Explanation

We want to find the number of ways a man can enter and exit a room through 10 distinct doors, with the condition that he must exit through a door different from the one he entered.

1. **Entering the Room:**
The man can choose to enter the room through any of the 10 available doors.
Ways to enter=10 ways\text{Ways to enter} = 10 \text{ ways}

2. **Exiting the Room:**
Since the man must exit through a *different* door, the door he used to enter cannot be used for exiting. This leaves:
10−1=9 available doors for exiting.10 - 1 = 9 \text{ available doors for exiting.}
Ways to exit=9 ways\text{Ways to exit} = 9 \text{ ways}

By the fundamental multiplication principle of counting, the total number of ways to complete the entire journey is:
Total ways=Ways to enter×Ways to exit=10×9=90\text{Total ways} = \text{Ways to enter} \times \text{Ways to exit} = 10 \times 9 = 90

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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