Permutations and CombinationsMCQMTP May 20Question 1664 of 251
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An examination paper with 10 questions consists of 6 questions in Algebra and 4 questions in Geometry. At least one question from each section is to be attempted. In how many ways can this be done?

Options

A945
B100
C1000
Dnone of these
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Correct Answer

✅ Option a — 945

All Options:

  • A945
  • B100
  • C1000
  • Dnone of these

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Detailed Solution & Explanation

We are given a question paper with two sections:
1. Algebra section containing 6 questions.
2. Geometry section containing 4 questions.

A student must attempt at least one question from each section. Let us calculate the number of ways to make selections for each section independently.

For the Algebra section containing 6 distinct questions:
Each of the 6 questions can either be attempted or not attempted (2 choices for each question).
The total number of choices is 26=64\displaystyle 2^6 = 64.
Since the student must attempt at least one Algebra question, we exclude the single case where no Algebra questions are attempted (0 questions selected):
Ways to attempt Algebra=26−1=64−1=63\text{Ways to attempt Algebra} = 2^6 - 1 = 64 - 1 = 63

For the Geometry section containing 4 distinct questions:
Similarly, each of the 4 questions can either be attempted or not attempted, yielding 24=16\displaystyle 2^4 = 16 choices.
Since the student must attempt at least one Geometry question, we exclude the case where no Geometry questions are attempted:
Ways to attempt Geometry=24−1=16−1=15\text{Ways to attempt Geometry} = 2^4 - 1 = 16 - 1 = 15

Using the fundamental multiplication principle of counting, the total number of ways to attempt the examination paper is:
Total ways=Ways to attempt Algebra×Ways to attempt Geometry\text{Total ways} = \text{Ways to attempt Algebra} \times \text{Ways to attempt Geometry}
Total ways=63×15=945\text{Total ways} = 63 \times 15 = 945

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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