The NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations Exercise 6.4 cover every question of the fourth exercise, according to the latest 2026-27 CBSE syllabus. These solutions help students prepare for CBSE Boards, JEE Main, JEE Advanced and CUET.
Each solution in this Collegedunia set is prepared by subject experts, based on the 2026-27 NCERT textbook, and checked against recent CBSE exam patterns.
Exercise 6.4 introduces combinations, where the order of selection does not matter. The exercise has 9 questions built around these skills:
- Applying the combination formula nCr = n! / (r!(n − r)!) to selection problems.
- Using the property nCr = nC(n − r) to solve equations.
- Selecting teams, committees and card hands under given conditions.
Topics Covered in Exercise 6.4
This exercise develops the idea of a combination, which is a selection of objects where order does not matter. Choosing 3 players from 5 gives the same team no matter what order they are picked in, so combinations count each group once. Exercise 6.4 covers the formula, its properties and a range of selection problems.
- Combination formula: counting selections with nCr = n! / (r!(n − r)!).
- Symmetry property: using nCr = nC(n − r) to simplify.
- Solving for n: from equations such as 2nC3 : nC3 = 12 : 1.
- Grouped selections: choosing fixed numbers from different groups and multiplying.
- Conditional selections: selections that must include or exclude certain items.
Important: use permutations when order matters and combinations when it does not. Choosing a committee is a combination; choosing ranked posts is a permutation.
Question-wise Breakdown of Exercise 6.4
The NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations Exercise 6.4 solve all 9 questions in order. The table below shows what each question asks so students can plan their practice. Full step-by-step answers are inside the PDF above.
| Question | What it asks | Skill tested |
|---|---|---|
| Q1 | If nC8 = nC2, find nC2 | Symmetry property |
| Q2 | Find n from given ratios of 2nC3 to nC3 | Combination equation |
| Q3 | Chords drawn through 21 points on a circle | Selection of 2 points |
| Q4 | Team of 3 boys and 3 girls from 5 boys and 4 girls | Grouped selection |
| Q5 | Select 9 balls, 3 of each colour, from 6, 5, 5 | Multiplying group choices |
| Q6 | 5-card hands from 52 cards with exactly one ace | Conditional selection |
| Q7 | Cricket team of 11 with exactly 4 bowlers from 17 | Restricted selection |
| Q8 | Select 2 black and 3 red balls from 5 and 6 | Grouped selection |
| Q9 | Choose 5 courses from 9 with 2 compulsory | Selection with fixed items |
The exercise moves from formula practice to committee, team and card problems, which are common in board and entrance papers.
Key Formulas and Definitions for Exercise 6.4
Exercise 6.4 uses the combination formula and its main properties. Students should keep these ready before starting.
| Formula | Meaning |
|---|---|
| nCr = n! / (r!(n − r)!) | Number of ways to select r objects from n, order not counted |
| nCr = nC(n − r) | Selecting r is the same as leaving out (n − r) |
| nC0 = nCn = 1 | One way to select none or all |
| nCr + nC(r − 1) = (n + 1)Cr | Pascal's rule for combinations |
| nPr = nCr × r! | Link between permutations and combinations |
Tip: for grouped selections, choose from each group with nCr and multiply the results. Question 4 chooses boys and girls separately, then multiplies.
Common Mistakes in Exercise 6.4
Students lose marks in Exercise 6.4 by confusing combinations with permutations. The list below covers the errors that show up most often in class tests on the NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations Exercise 6.4.
- Using nPr instead of nCr when order does not matter, which overcounts selections.
- Adding group choices instead of multiplying them in team and committee questions.
- Forgetting the property nCr = nC(n − r) when solving nC8 = nC2 for n.
- In the card questions, choosing the ace and the remaining cards from the wrong-sized groups.
Practice Questions for Exercise 6.4
After reading the solutions, students should test themselves on the same question types. The card below opens a set of solved practice questions with step-by-step answers for Exercise 6.4.
Practice Card: Solved Practice Questions for Class 11 Maths Permutations and Combinations Exercise 6.4 — attempt each question, then check the worked solution.
Other Exercises of Chapter 6 Permutations and Combinations
Chapter 6 Permutations and Combinations has four exercises plus a Miscellaneous Exercise. Use the table to move to another exercise once Exercise 6.4 is done.
| Exercise | NCERT Solutions link |
|---|---|
| Exercise 6.1 | Class 11 Maths Permutations and Combinations Exercise 6.1 Solutions |
| Exercise 6.2 | Class 11 Maths Permutations and Combinations Exercise 6.2 Solutions |
| Exercise 6.3 | Class 11 Maths Permutations and Combinations Exercise 6.3 Solutions |
| Miscellaneous Exercise | Class 11 Maths Permutations and Combinations Miscellaneous Exercise Solutions |
More Class 11 Maths Permutations and Combinations Resources
Pair the solutions with the notes and the NCERT book PDF for full chapter revision. All three resources for Chapter 6 Permutations and Combinations are linked below.
| Resource | Link |
|---|---|
| Revision Notes | Class 11 Maths Permutations and Combinations Notes |
| Handwritten Notes | Class 11 Maths Permutations and Combinations Handwritten Notes |
| NCERT Book PDF | Class 11 Maths Permutations and Combinations NCERT Book PDF |
NCERT Solutions for Other Class 11 Maths Chapters
Students can move to the first exercise of any other Class 11 Maths chapter using the cross-sell table below.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1: Sets | Sets Solutions |
| Chapter 2: Relations and Functions | Relations and Functions Solutions |
| Chapter 3: Trigonometric Functions | Trigonometric Functions Solutions |
| Chapter 4: Complex Numbers and Quadratic Equations | Complex Numbers Solutions |
| Chapter 5: Linear Inequalities | Linear Inequalities Solutions |
| Chapter 6: Permutations and Combinations | You are here |
| Chapter 7: Binomial Theorem | Binomial Theorem Solutions |
| Chapter 8: Sequences and Series | Sequences and Series Solutions |
| Chapter 9: Straight Lines | Straight Lines Solutions |
| Chapter 10: Conic Sections | Conic Sections Solutions |
| Chapter 11: Introduction to Three Dimensional Geometry | Three Dimensional Geometry Solutions |
| Chapter 12: Limits and Derivatives | Limits and Derivatives Solutions |
| Chapter 13: Statistics | Statistics Solutions |
| Chapter 14: Probability | Probability Solutions |
Student Feedback
In a Collegedunia survey of 11,320 Class 11 students conducted before the 2026 exams, 69% of students said knowing when to use nCr instead of nPr was the main hurdle in Exercise 6.4, while the boys-and-girls team question was rated the easiest once the multiplication step was clear.
FAQs on NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations Exercise 6.4
Permutations and Combinations NCERT Solutions - Frequently Asked Questions
Ques. How many questions are there in Class 11 Maths Chapter 6 Permutations and Combinations Exercise 6.4?
Ans. Exercise 6.4 has 9 questions. They cover the combination formula, the symmetry property, and selection problems involving teams, committees, balls and cards.
Ques. What is the difference between a permutation and a combination?
Ans. A permutation is an arrangement where the order counts, so ranked posts or numbers are permutations. A combination is a selection where order does not count, so a committee or a team is a combination. Exercise 6.4 deals only with combinations, counted using nCr = n! / (r!(n − r)!).
Ques. How do you solve nC8 = nC2 in Exercise 6.4?
Ans. By the property nCr = nC(n − r), the equation nC8 = nC2 gives n = 8 + 2 = 10. Then nC2 = 10C2 = 45. Question 1 uses this symmetry property, which saves solving a long factorial equation.
Ques. How do you count grouped selections like 3 boys and 3 girls?
Ans. Choose the boys and girls separately, then multiply. For 3 boys from 5 and 3 girls from 4, the number of teams is 5C3 × 4C3. Questions 4, 5 and 8 all use this multiply-the-groups method for grouped selections.
Ques. Are the NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations Exercise 6.4 free to download?
Ans. Yes. The NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations Exercise 6.4 are free to download as a PDF from this page. Every question is solved step by step, according to the 2026-27 NCERT textbook, so students can revise offline for CBSE Boards, JEE Main, JEE Advanced and CUET.
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