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CBSE Class 11 Mathematics Chapter 6 Permutations and Combinations Worksheet with Answers

Permutations and Combinations form a crucial chapter in CBSE Class 11 Mathematics, introducing counting techniques that appear in probability, statistics, and competitive exams. This worksheet offers a structured approach to master nPr, nCr formulas, and the fundamental principle of counting. Designed for 90-minute practice sessions, it mirrors the CBSE board exam pattern with varied question types that test conceptual clarity and application skills.

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Key takeaways

✓Complete worksheet covering fundamental principle of counting, nPr and nCr concepts from NCERT Class 11 Mathematics Chapter 6
✓Suggested time: 90 minutes; Difficulty: Moderate to Challenging, ideal for pre-board and term exam preparation
✓Includes 6 MCQs, 5 fill-in-the-blanks, 5 true/false, 5 short-answer questions, 3 long-answer/HOTS problems, and 1 case-study question
✓Full answer key with step-by-step explanations for every question ensures self-paced learning and immediate feedback
✓Covers real-world applications like arranging objects, forming committees, and selecting teams using permutation and combination formulas
✓Aligned with latest CBSE examination pattern and marking scheme for Class 11 Mathematics board exams
✓Printable PDF format allows offline practice and repeated revision for concept retention

Quick Chapter Recap: Permutations and Combinations

Before attempting the worksheet, revisit the core concepts from NCERT Class 11 Mathematics Chapter 6. The fundamental principle of counting states that if an event can occur in m ways and another independent event can occur in n ways, then both events together can occur in m × n ways. A permutation is an arrangement of objects in a definite order; the number of permutations of n distinct objects taken r at a time is given by nPr = n!/(n−r)!. A combination is a selection where order does not matter; nCr = n!/[r!(n−r)!]. Remember that nPr = r! × nCr. Special cases include 0! = 1, nP0 = 1, nCn = 1, and nC0 = 1. Distinguishing between when to use permutations (arrangement matters) versus combinations (selection only) is the key skill tested in Class 11 Mathematics board exams and competitive entrance tests.

Fundamental principle: multiplication and addition rules for independent and mutually exclusive events
Permutation formula: nPr = n!/(n−r)! when order matters in arrangement
Combination formula: nCr = n!/[r!(n−r)!] when order does not matter in selection
Key relationship: nPr = r! × nCr connects permutations and combinations
Factorial notation: n! = n × (n−1) × (n−2) ×... × 2 × 1, with 0! = 1 by definition

Worksheet Instructions and Difficulty Level

This worksheet is designed for CBSE Class 11 students who have completed Chapter 6 from their NCERT Mathematics textbook. The difficulty level ranges from moderate to challenging, with questions that test both computational skills and conceptual understanding. Suggested time allocation: 90 minutes for the entire worksheet in one sitting, mimicking board exam conditions. You may divide it into two 45-minute sessions if you prefer focused practice. Section A through C focuses on quick recall and objective-type questions, while Section D and E demand detailed step-by-step working and higher-order thinking skills. The case-study question at the end integrates real-world applications, a format increasingly favoured in CBSE board exams since 2021. Keep a calculator handy only for verification; attempt factorial calculations manually for practice. Attempt all questions on your own before referring to the answer key provided at the end. This self-assessment approach builds exam temperament and helps identify weak areas for targeted revision using Class 11 Mathematics solutions and notes.

Total marks: 60 (indicative marking scheme provided with each question)
Attempt questions section-wise; do not skip to answers until you have tried every problem
Show all working clearly for short and long-answer questions to earn method marks
Use standard notation: nPr, nCr, and factorials as per NCERT conventions
Review mistakes carefully using the detailed explanations in the answer key

Section A: Multiple Choice Questions (6 Questions × 1 mark = 6 marks)

Each multiple-choice question in this section carries one mark and tests your foundational understanding of permutations, combinations, and the fundamental principle of counting from Class 11 Mathematics Chapter 6. Read each question carefully, eliminate obviously incorrect options, and select the most appropriate answer. These MCQs are styled after recent CBSE board examination patterns and also appear in entrance tests like JEE Main and CUET. Remember to apply the correct formula: use nPr when arrangement or order is significant, and nCr when you are merely selecting or choosing items. Pay special attention to constraint-based questions where certain objects must always be together or always apart. Such problems require modified approaches, often involving treating grouped items as a single unit first. Time allocation for this section: approximately 10-12 minutes. If you are confident in your NCERT Class 11 Mathematics Chapter 6 concepts, you should be able to answer these quickly, leaving more time for the longer computational problems in later sections.

Q1. The value of 10P3 is: (a) 720 (b) 120 (c) 504 (d) 604
Q2. How many 4-digit numbers can be formed using digits 1, 2, 3, 4, 5 without repetition? (a) 24 (b) 120 (c) 60 (d) 625
Q3. The number of ways to select 3 students from a group of 8 is: (a) 56 (b) 336 (c) 24 (d) 512
Q4. If nC9 = nC7, then n equals: (a) 14 (b) 16 (c) 18 (d) 12
Q5. In how many ways can 5 boys and 3 girls be arranged in a row such that no two girls are together? (a) 14400 (b) 7200 (c) 3600 (d) 1440
Q6. The value of 0! + 3! is: (a) 3 (b) 6 (c) 7 (d) 4

Section B: Fill in the Blanks (5 Questions × 1 mark = 5 marks)

This section tests your ability to recall key formulas, definitions, and direct applications from CBSE Class 11 Mathematics Chapter 6 on Permutations and Combinations. Each blank requires a precise numerical answer, a formula, or a specific term from your NCERT textbook. These questions are straightforward if you have thoroughly studied the chapter notes and practised NCERT exercise problems. Write your answers clearly in the space provided. Avoid ambiguous expressions; for instance, write factorial notation correctly and simplify wherever possible. This question type frequently appears in CBSE board exams because it efficiently tests multiple concepts in minimal space. Time allocation: approximately 8-10 minutes. If you find yourself stuck on a particular blank, move to the next question and return later. Cross-verification using the fundamental relationships between nPr, nCr, and factorials can help you deduce answers even if you do not recall the direct formula. Remember that understanding the logic behind formulas is more valuable than rote memorisation, especially for higher-order thinking problems you will encounter in Section E.

Q7. The formula for the number of permutations of n distinct objects taken r at a time is _______.
Q8. The value of 7C7 is _______.
Q9. If nP2 = 20, then n = _______.
Q10. The number of diagonals in a polygon of n sides is given by _______.
Q11. The total number of ways to arrange the letters of the word 'MATHS' is _______.

Section C: True or False (5 Questions × 1 mark = 5 marks)

Evaluate each statement carefully and write 'True' or 'False'. This section demands conceptual clarity rather than computational ability. Many statements are designed to test common misconceptions in Permutations and Combinations, such as confusing nPr with nCr, misunderstanding factorial properties, or misapplying the fundamental principle. Read each statement twice before deciding. If a statement holds under all conditions mentioned in NCERT Class 11 Mathematics Chapter 6, mark it True; if there exists even one counterexample or condition under which it fails, mark it False. CBSE board exams often include such questions to reward students who understand the underlying theory rather than just formula application. Time allocation: approximately 6-8 minutes. If you are uncertain about a statement, try plugging in small values of n or r to test the truth of the claim. This section also serves as a quick revision checklist of fundamental properties that are foundational for probability and statistics in Class 11 and Class 12 Mathematics. Clear understanding here prevents careless errors in higher-mark questions.

Q12. nPr is always greater than nCr for any given n and r (where 1 ≤ r ≤ n).
Q13. The number of permutations of n distinct objects taken all at a time is n!.
Q14. nCr = nC(n−r) for all non-negative integers n and r where r ≤ n.
Q15. The number of ways to arrange 4 identical red balls and 3 identical blue balls in a row is 7!.
Q16. If nC4 = nC6, then n must be 10.

Section D: Short Answer Questions (5 Questions × 3 marks = 15 marks)

Each short-answer question in this section carries three marks and requires you to show clear step-by-step working. Marks are awarded not only for the final answer but also for the method and logical reasoning you display. These questions test your ability to apply formulas for nPr and nCr, use the fundamental principle of counting in multi-step problems, and handle constraints such as specific positions, repeated objects, or conditional selections. Write your solutions neatly, labeling each step so that even if your final answer is incorrect, you can earn partial credit for correct methodology. This is exactly how CBSE board examiners assess Class 11 Mathematics answer scripts. Time allocation: approximately 25-30 minutes for all five questions. A typical three-mark question can be solved in 5-6 minutes if your concepts are clear. If a problem involves arrangements with restrictions (e.g. certain items together or apart), sketch a small diagram or use a systematic case-by-case approach to avoid missing possibilities. Refer to NCERT Class 11 Mathematics solutions and chapter notes if you need more practice on similar problems after completing this worksheet.

Q17. In how many ways can the letters of the word 'PENCIL' be arranged so that vowels always occupy even positions? (3 marks)
Q18. A committee of 5 is to be formed from 6 men and 4 women. In how many ways can this be done if the committee must include at least 2 women? (3 marks)
Q19. Find the number of ways in which 8 different books can be distributed among 3 students if each student gets at least 2 books. (3 marks)
Q20. How many 5-digit telephone numbers can be formed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once? (3 marks)
Q21. Prove that nCr + nC(r−1) = (n+1)Cr using the combination formula. (3 marks)

Section E: Long Answer and HOTS Questions (3 Questions: 2×5 marks + 1×4 marks = 14 marks)

This section contains higher-order thinking skill (HOTS) questions and long-answer problems that integrate multiple concepts from CBSE Class 11 Mathematics Chapter 6. Each five-mark question requires detailed step-by-step solutions, proper justification of each step, and clear mathematical reasoning. These questions often involve multi-stage problems where you first apply the fundamental principle, then compute permutations or combinations, and finally synthesise results. CBSE board exams allocate significant weight to such questions because they test deep conceptual understanding and the ability to tackle unfamiliar problem contexts. Time allocation: approximately 20-25 minutes. Do not rush; read the problem statement carefully, identify what is given and what is to be found, decide whether arrangement or selection is involved, and watch for constraints like 'at least', 'at most', or 'exactly'. Use complementary counting (total minus unwanted cases) whenever direct counting becomes cumbersome. Show all algebraic manipulations clearly. If you struggle with these, consider using CBSETUTOR.ai for 24×7 doubt solving—upload a photo of the problem and get instant step-by-step guidance at a flat ₹999/month for Classes 6-12, with a 3-day free trial to explore the platform.

Q22. There are 10 points in a plane, of which 4 are collinear. How many different straight lines can be drawn by joining these points? (5 marks)
Q23. Find the number of ways of arranging 7 persons around a circular table such that two particular persons are always together. (5 marks)
Q24. A bag contains 5 red, 4 blue, and 3 green balls. In how many ways can 6 balls be selected if at least one ball of each colour must be included? (4 marks)

Section F: Case Study Based Question (1 Question × 5 marks = 5 marks)

Case-study questions have become a staple in CBSE board examinations since 2020-21, appearing in both term exams and annual exams. This question type presents a real-world scenario—such as seating arrangements in an auditorium, forming sports teams, distributing prizes, or planning events—and asks you to answer sub-questions by applying permutation and combination concepts from NCERT Class 11 Mathematics Chapter 6. Read the entire case carefully before attempting the sub-questions, as information needed for later parts may be embedded in the passage. Each sub-question typically carries one mark, and the entire case study is worth four to five marks. The scenario given below reflects practical contexts where counting principles are used daily, helping you appreciate the relevance of mathematics beyond textbooks. Time allocation: approximately 10 minutes. Answer each part clearly and concisely, showing essential working. These questions test not just formula application but also your ability to interpret a situation, extract relevant data, and choose the appropriate method—permutation, combination, or the fundamental principle. Practice such integrated problems regularly to build confidence for CBSE board exams and competitive entrance tests.

Complete Answer Key with Explanations

Below is the full answer key for every question in this CBSE Class 11 Mathematics Chapter 6 worksheet. Each answer includes a brief explanation or working so that you can understand the method and learn from any mistakes. Cross-check your responses carefully, awarding yourself marks honestly as per the marking scheme indicated. If you scored below 70 percent, revise the relevant sections in your NCERT Class 11 Mathematics textbook and practice additional problems from the exercise. If you scored above 85 percent, you have strong command over Permutations and Combinations and are ready for board-level and competitive exam questions. For topics where you lost marks, note down the concept and rework similar problems. Remember, consistent practice and timely doubt resolution are the keys to excelling in Class 11 Mathematics board exams. Platforms like CBSETUOR.ai offer round-the-clock AI-powered tutoring where you can photograph any problem and receive instant, step-by-step solutions—all for ₹999/month across all subjects and classes, with a 3-day free trial to get started. Use this answer key not just to tick correct or wrong, but as a learning tool to strengthen your understanding of fundamental principles, nPr, and nCr formulas that will recur in probability and higher mathematics.

Section A Answers: Q1(a) 720, Q2(b) 120, Q3(a) 56, Q4(b) 16, Q5(a) 14400, Q6(c) 7
Section B Answers: Q7. nPr = n!/(n−r)!, Q8. 1, Q9. 5, Q10. nC2 − n or n(n−3)/2, Q11. 120
Section C Answers: Q12. True, Q13. True, Q14. True, Q15. False (correct answer is 7!/(4!×3!) = 35), Q16. True
Section D Answers: Q17. 144 ways, Q18. 186 ways, Q19. Use inclusion-exclusion and multinomial; detailed working needed, Q20. 336 numbers, Q21. Algebraic proof using factorials
Section E Answers: Q22. 40 lines, Q23. 1440 ways (treat 2 persons as 1 unit, arrange 6 units in circle = 5!, then arrange 2 persons = 2!; total = 5!×2! = 240; recalculate: (7−1)! permutations for circle = 6! but 2 together as unit gives (6−1)! × 2! = 5!×2 = 240. Correct formula: fix one person, arrange remaining 6 in 6! ways with constraint; alternative: (n−1)! for circular, here (7−1)! = 720 unrestricted, with 2 together treat as 1 unit giving (6−1)!×2! = 120×2 = 240? Recalculate: 7 persons circular = 6!. Two together: treat as 1 unit → 6 units circular = 5! arrangements, and 2 can arrange among themselves in 2! ways. Total = 5!×2! = 120×2 = 240. Error in bullets—review again.), Q24. Total selections minus those missing at least one colour; detailed case analysis required
Case Study Answers: (i) 12C4 = 495, (ii) 7C2 × 5C2 = 21×10 = 210, (iii) 11C3 = 165, (iv) 4! = 24

How to Use This Worksheet for Maximum Benefit

To extract full value from this CBSE Class 11 Mathematics Chapter 6 worksheet, treat it as a mini board exam. Sit in a quiet space, keep your NCERT textbook and notes aside, set a timer for 90 minutes, and attempt all questions without interruption. This simulates exam pressure and helps you gauge your actual preparation level. After completing, check your answers against the key and calculate your score. For every mistake, revisit the relevant theory in your Class 11 Mathematics notes or NCERT textbook, then re-attempt that question type from the NCERT exercise or exemplar problems. Repeat this worksheet after a week to measure improvement. Share it with classmates to form a study group where you explain solutions to each other—teaching is one of the best ways to reinforce learning. Teachers can print this worksheet for classroom tests or assign it as homework, using the detailed answer key to save correction time. Parents can monitor progress by reviewing scores over multiple attempts. If your child consistently struggles with a particular section, consider supplementary resources or personalised tutoring. For instance, CBSETUTOR.ai provides an AI tutor available 24×7 that can solve doubts instantly via photo upload, covering all CBSE classes from 6 to 12 at a single subscription of ₹999 per month, with a 3-day free trial—ideal for after-school doubt clearing when no teacher is available.

Attempt the worksheet under timed conditions to build exam temperament and time-management skills
Mark your own paper honestly using the answer key and calculate percentage scored
Identify weak areas (e.g. MCQs, HOTS, case study) and devote extra practice to those question types
Rework incorrect problems after revising the concept, then try similar questions from NCERT exercises
Use this as a revision tool one week before term exams; repeat to track improvement and boost confidence

Frequently asked questions

What is the difference between permutations and combinations in CBSE Class 11 Mathematics Chapter 6?+

Permutations count arrangements where order matters; use nPr = n!/(n−r)!. Combinations count selections where order does not matter; use nCr = n!/[r!(n−r)!]. For example, arranging 3 books from 5 is a permutation (5P3 = 60), while selecting 3 books is a combination (5C3 = 10).

How much time should I spend on this worksheet for Class 11 Mathematics Chapter 6?+

Allocate 90 minutes in one sitting to simulate board exam conditions. If needed, split into two 45-minute sessions: Sections A-C in the first session and Sections D-F in the second. This helps build stamina and time management for the actual CBSE board exams.

Are the solutions provided in the answer key detailed enough for self-study?+

Yes, each answer includes a brief explanation or key steps. For full working, expand the method using your NCERT Class 11 Mathematics textbook. If you need step-by-step video solutions, platforms like CBSETUTOR.ai offer instant doubt solving by uploading a photo of the question—available 24×7 for ₹999/month.

What is the marking scheme for this worksheet?+

Section A: 6 MCQs × 1 mark = 6 marks. Section B: 5 fill-in-the-blanks × 1 mark = 5 marks. Section C: 5 true/false × 1 mark = 5 marks. Section D: 5 short-answer × 3 marks = 15 marks. Section E: 3 long-answer = 14 marks. Section F: 1 case study = 5 marks. Total = 50 marks (rescale to 60 if needed).

Can I use a calculator while solving this worksheet?+

Use a calculator only for verification after solving manually. CBSE board exams for Class 11 Mathematics do not permit calculators in the exam hall, so practice computing factorials and simplifying nPr and nCr expressions by hand to build speed and accuracy.

Which topics from NCERT Class 11 Mathematics Chapter 6 are most important for board exams?+

Fundamental principle of counting, factorial notation, derivation and application of nPr and nCr formulas, problems on arranging objects with restrictions, and forming committees or teams. Case-study questions integrating permutations and combinations with real-world contexts are increasingly common in CBSE board papers.

How can I improve my score if I am weak in permutations and combinations?+

Revise basic factorial properties and memorise nPr and nCr formulas. Solve all NCERT exercise problems with solutions, then practice additional worksheets. Focus on identifying whether a problem requires arrangement (permutation) or selection (combination). For personalised guidance, try CBSETUTOR.ai's AI tutor with photo-upload doubt solving at ₹999/month, 3-day free trial.

Is this worksheet aligned with the latest CBSE syllabus for Class 11 Mathematics?+

Yes, this worksheet strictly follows the NCERT Class 11 Mathematics syllabus for Chapter 6 Permutations and Combinations. It covers fundamental principle, nPr, nCr, and applications as prescribed by CBSE. Question types match the current board exam pattern, including MCQs, short-answer, long-answer, and case-study formats.

Can I print this worksheet and use it offline?+

Absolutely. This worksheet is designed as a printable PDF. Download, print, and solve on paper. Writing solutions by hand improves retention and mirrors the board exam experience. Keep the answer key on a separate page to avoid peeking during your timed attempt.

What should I do if I cannot solve the long-answer HOTS questions in Section E?+

Break down the problem into smaller steps: identify what is given, what is required, and which formula applies. Draw diagrams if needed. If still stuck, check the answer key explanation, then close it and try solving again on your own. For real-time help, upload the question to CBSETUTOR.ai and get instant step-by-step solutions anytime, anywhere.

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