NCERT Solutions Class 11 Maths Chapter 6 teaches you how to solve counting problems using permutations (arrangements) and combinations (selections). You’ll learn the fundamental principle of counting, derive formulas like nPr and nCr, and apply them to real-world problems involving arrangements of objects, formation of committees, and probability foundations. These problem-solving techniques are crucial for competitive exams and form the basis for statistics and probability in Class 12.
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All exercises with step-by-step solutions | Updated 2025-26 | Free Download
Download PDF (Free)NCERT Solutions Class 11 Maths Chapter 6 Permutations and Combinations – Complete Guide
NCERT Class 11 Chapter 6 on Permutations and Combinations introduces you to the fascinating world of counting principles that form the backbone of probability theory and discrete mathematics. You’ll explore the Fundamental Principle of Counting, which provides a systematic approach to determining the number of possible outcomes in various situations. This chapter builds your problem-solving skills through factorial notation, permutations (arrangements where order matters), and combinations (selections where order doesn’t matter).
📊 CBSE Class 11 Maths Chapter 6 – Exam Weightage & Marking Scheme
| CBSE Board Marks | 5 Marks |
| Unit Name | Algebra |
| Difficulty Level | Medium |
| Importance | Medium |
| Exam Types | CBSE Board, State Boards |
| Typical Questions | 1-2 questions |
You will learn to distinguish between situations requiring permutations versus combinations, a critical skill for CBSE board examinations where this chapter typically carries 5 marks. The problems range from simple counting exercises to complex real-world applications involving restrictions and conditions. You’ll work with formulas like nPr and nCr, understanding when and how to apply them effectively. This chapter also introduces important concepts like permutations with repetitions and circular permutations.
The exam questions from this chapter usually include 2-3 mark MCQs and short answer questions, along with occasional 4-mark application-based problems. Understanding permutations and combinations is crucial not just for Class 11, but also for Class 12 probability chapters and competitive exams like JEE and NEET. The concepts you learn here have practical applications in computer science, cryptography, and statistical analysis.
Quick Facts – Class 11 Chapter 6
| 📖 Chapter Number | Chapter 6 |
| 📚 Chapter Name | Permutations and Combinations |
| ✏️ Total Exercises | 4 Exercises |
| ❓ Total Questions | 31 Questions |
| 📅 Updated For | CBSE Session 2025-26 |
Mastering this chapter requires consistent practice with diverse problem types. Focus on understanding the logic behind each formula rather than rote memorization. Work through the NCERT exercises systematically, paying special attention to problems involving restrictions and special cases. With dedicated practice, you’ll develop the analytical thinking needed to tackle any counting problem confidently in your CBSE board exams.
NCERT Solutions Class 11 Maths Chapter 6 – All Exercises PDF Download
Download exercise-wise NCERT Solutions PDFs for offline study
| Exercise No. | Topics Covered | Download PDF |
|---|---|---|
| EXERCISE 6.1 | Complete step-by-step solutions for 6 questions | 📥 Download PDF |
| EXERCISE 6.2 | Complete step-by-step solutions for 5 questions | 📥 Download PDF |
| EXERCISE 6.3 | Complete step-by-step solutions for 11 questions | 📥 Download PDF |
| EXERCISE 6.4 | Complete step-by-step solutions for 9 questions | 📥 Download PDF |
Permutations and Combinations – Key Formulas & Concepts
Quick reference for CBSE exams
| Formula | Description | When to Use |
|---|---|---|
| Fundamental Principle of Counting (Multiplication) If one operation can be performed in \(m\) ways and another independent operation can be performed in \(n\) ways, then the two operations can be performed together in \(m \times n\) ways. | Calculates the total number of ways to perform multiple independent tasks. Note: Ensure the tasks are independent. If one task affects the other, this principle may not directly apply. | When you have multiple steps in a task and need the total possibilities (e.g., choosing an outfit with different shirts and pants) |
| Fundamental Principle of Counting (Addition) If one operation can be performed in \(m\) ways and another operation can be performed in \(n\) ways, and the two operations cannot be performed simultaneously, then either of the two operations can be performed in \(m + n\) ways. | Calculates the total number of ways to perform one task OR another (mutually exclusive). Note: Make sure the tasks are mutually exclusive (cannot happen at the same time). Avoid double-counting. | When you have options to do one thing OR another, but not both at the same time (e.g., choosing a fruit from either apples OR oranges) |
| Permutation (nPr) \(^nP_r = \frac{n!}{(n-r)!}\) | Number of ways to arrange \(r\) objects from a set of \(n\) distinct objects, where order matters. Note: Remember \(n!\) means n factorial (n * (n-1) * (n-2) * … * 1). \(r\) must be less than or equal to \(n\). | When the order of selection is important (e.g., arranging books on a shelf, ranking players). |
| Permutation with Repetition Number of permutations of \(n\) objects where \(p_1\) are alike of one kind, \(p_2\) are alike of another kind, …, \(p_k\) are alike of \(k\)th kind is \(\frac{n!}{p_1! p_2! … p_k!}\) | Calculates arrangements when some objects are identical. Note: Make sure to divide by the factorial of the number of repetitions for each type of object. | When you have repeating elements and need to find distinct arrangements (e.g., arranging letters in the word ‘MISSISSIPPI’). |
| Combination (nCr) \(^nC_r = \frac{n!}{r!(n-r)!}\) | Number of ways to choose \(r\) objects from a set of \(n\) distinct objects, where order does NOT matter. Note: Remember that \(\^nC_r = \^nC_{n-r}\). This can simplify calculations. | When the order of selection is NOT important (e.g., forming a committee, selecting lottery numbers). |
| Combination with Repetition Allowed \(^{n+r-1}C_r = \frac{(n+r-1)!}{r!(n-1)!}\) | Number of ways to choose \(r\) objects from \(n\) distinct objects with repetition allowed. Note: This is a less common but important formula. Make sure you understand when repetition is allowed. | When you can choose the same object multiple times (e.g., selecting ice cream flavors when you can have multiple scoops of the same flavor). |
| Number of ways to divide (m+n) distinct objects into two groups containing m and n objects \( \frac{(m+n)!}{m!n!} \) | Divide (m+n) distinct objects into two groups containing m and n objects Note: Similar to combinations but focuses on dividing all items | Dividing people into teams, distributing items into different groups. |
| Circular Permutations \((n-1)!\) | Number of ways to arrange \(n\) distinct objects in a circle. Note: In a circle, there’s no fixed starting point. Therefore, we fix one object and arrange the rest. | When arranging people around a circular table or objects in a ring. |
| Restricted Permutations Depends on the specific problem. Consider fixing certain elements or excluding others. | Permutations with specific conditions (e.g., certain objects must be together or never together). Note: These problems require careful reading and often involve subtracting unwanted arrangements from the total possible arrangements. | When a problem states restrictions on how objects can be arranged. |
| Restricted Combinations Depends on the specific problem. Consider including or excluding certain elements. | Combinations with specific conditions (e.g., certain objects must be included or excluded). Note: Similar to restricted permutations, these problems require careful analysis and may involve using the principle of inclusion-exclusion. | When a problem states restrictions on how objects can be selected. |
Frequently Asked Questions – NCERT Class 11 Maths Chapter 6
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