NCERT Solutions for Class 11 Maths Chapter 7 – Permutations and Combinations
Of all the chapters in Class 11 Mathematics, Chapter 7 is the one that students find simultaneously the most exciting and the most confusing. The mathematics here is about counting — not just how many, but how many different ways. And the answer almost always depends on one critical question: does the order of selection matter?
Permutations deal with arrangements, where order matters. Combinations deal with selections, where order does not. Knowing which one applies to a given problem is the central skill this chapter builds. Get that wrong, and your answer will be off by a factor of n! — which can be very large. For all Chapters must, read NCERT Solutions for Class 11 Maths and subject-wise NCERT Solutions for Class 11.
NCERT builds this chapter carefully. It starts with the Fundamental Principle of Counting (Exercise 7.1), moves to factorials and permutations (Exercise 7.2 and 7.3), and culminates in combinations (Exercise 7.4). The miscellaneous exercise mixes all of these and adds real-world counting problems involving words formed from letters, teams selected from groups, and paths through grids. The NCERT solutions here show the full reasoning for why a particular formula is applied, not just the calculation — because in PnC problems, setting up is harder than computing.
Download PDF – All Exercises of NCERT Solutions for Class 11 Maths Chapter 7 Permutations and Combination
| Exercise | Topic Covered | Number of Questions |
|---|---|---|
| Exercise 7.1 | Fundamental Principle of Counting | 6 Questions |
| Exercise 7.2 | Factorials | 5 Questions |
| Exercise 7.3 | Permutations (nPr) | 11 Questions |
| Exercise 7.4 | Combinations (nCr) | 9 Questions |
| Miscellaneous | Word Problems, Arrangements, Selections | 11 Questions |
Chapter 7 – Permutations and Combinations: Concepts, Explanation and Key Tables
The Fundamental Principle of Counting
If one event can occur in m ways and a second event can occur in n ways (independent of the first), then both events together can occur in m × n ways. This is the multiplication principle and it is the backbone of everything in this chapter.
Its counterpart — the addition principle — says: if event A can occur in m ways and event B in n ways, and the two events cannot occur simultaneously, then either A or B can occur in (m + n) ways.
Permutations – Arrangement Counts
A permutation is an arrangement of objects where the order of selection matters. The number of permutations of n distinct objects taken r at a time is:
ⁿPᵣ = n! / (n – r)!
| Formula | When to Use |
|---|---|
| ⁿPₙ = n! | Arranging all n objects |
| ⁿPᵣ = n!/(n–r)! | Selecting and arranging r from n distinct objects |
| n!/(p! × q! × r!) | Arrangements of n objects with p identical of one type, q of another, r of another |
| (n–1)! | Circular arrangements of n distinct objects |
| (n–1)! / 2 | Circular arrangements where clockwise and anticlockwise are the same (e.g. necklace) |
Combinations – Selection Counts
A combination is a selection of objects where the order does not matter.
ⁿCᵣ = n! / [r! × (n–r)!]
| Property | Statement |
|---|---|
| ⁿCᵣ = ⁿCₙ₋ᵣ | Selecting r from n is same as leaving out n–r |
| ⁿC₀ = ⁿCₙ = 1 | Selecting none or all = one way |
| ⁿC₁ = n | Selecting one item = n choices |
| ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ | Pascal's Identity (used in Binomial Theorem) |
| Sum of all ⁿCᵣ = 2ⁿ | Total number of subsets of an n-element set |
Permutation vs Combination — Decision Table
| Situation | Permutation or Combination? | Reasoning |
|---|---|---|
| Arranging letters in a word | Permutation | ABCD ≠ DCBA |
| Selecting a committee of 3 from 10 people | Combination | Order of selection irrelevant |
| Assigning 3 different prizes to students | Permutation | 1st, 2nd, 3rd are distinct |
| Choosing 2 toppings from 5 for a pizza | Combination | Tomato+Cheese = Cheese+Tomato |
| Number of 4-digit numbers from {1,2,3,4,5} | Permutation | 1234 ≠ 4321 |
| Number of handshakes in a group of n people | Combination | A shaking B's hand = B shaking A's hand |
| Seating arrangement at a round table | Circular Permutation | Relative order matters |
| Selecting a President, VP, and Secretary | Permutation | Roles are distinct |
Common Problems and Their Setup
| Problem Type | Setup | Formula Used |
|---|---|---|
| Words from letters of a word with no repetition | All letters distinct: n! / repetitions! | Permutation with identical objects |
| Teams with specific conditions (at least one woman) | Total – (no women) | Combination with complementary counting |
| Paths in a grid from (0,0) to (m,n) | m right moves + n up moves; total = m+n moves | (m+n)! / (m! × n!) = ⁽ᵐ⁺ⁿ⁾Cₘ |
| Selecting from two separate groups | Multiply selections from each group | Product of combinations |
| Number with digits from given set | Fill each place using permutation | ⁿPᵣ or positional counting |
Study Tips for Chapter 7
- Before applying any formula, ask: "Is the order of my selection/arrangement mattering here?" — Yes → Permutation, No → Combination.
- For word formation problems, list the distinct letters and their frequencies first before setting up the formula.
- Circular permutation problems must specify whether the arrangement is on a fixed circle (like seats) or a symmetrical one (like a necklace with identical sides) — the formula differs.
- The complementary counting technique (Total – Unfavourable) saves enormous calculation time in difficult combination problems and should be your first strategy when conditions are restrictive.
FAQs on NCERT Solutions Class 11 Maths Chapter 7 Permutations & Combinations
The main difference between permutations and combinations is the importance of order. In permutations, the arrangement of objects matters, meaning different orders are counted as separate outcomes. In combinations, only the selection matters, and the order of objects is not considered. For example, selecting two students from a group is a combination, while assigning first and second positions among students is a permutation. NCERT Solutions explain this difference through numerous examples and solved questions. Understanding when to use permutation formulas and when to apply combination formulas is essential for solving counting problems correctly and efficiently in examinations.
Factorial notation is the foundation of most formulas used in permutations and combinations. The factorial of a number, represented by n!, is the product of all positive integers from 1 to n. It helps calculate the number of possible arrangements and selections of objects. Students use factorials extensively while solving permutation and combination problems. NCERT Solutions provide step-by-step examples that demonstrate how factorial notation is applied in different situations. A strong understanding of factorials makes it easier to simplify formulas and solve complex counting questions. This concept is also important for probability and advanced mathematical topics studied later.
NCERT Solutions provide detailed and accurate explanations for every exercise question in the chapter. They help students understand counting principles, factorial notation, permutations, combinations, and their practical applications. Step-by-step solutions simplify complex problems and improve conceptual clarity. Students can use these solutions for classroom learning, homework completion, revision, and examination preparation. Regular practice enhances logical reasoning and mathematical accuracy. Since this chapter plays a significant role in probability and competitive examinations, mastering NCERT Solutions helps students build a strong foundation for higher-level mathematics and improves their confidence in solving challenging problems.
Students can identify the correct method by checking whether order matters in the given problem. If different arrangements create different outcomes, the question involves permutations. If only the selection of objects is important and the order does not matter, combinations should be used. For example, arranging books on a shelf requires permutations, while choosing a team from a group requires combinations. NCERT Solutions contain various examples that help students distinguish between these two concepts. Practicing different question types regularly develops the ability to recognize the correct approach quickly and improves problem-solving accuracy during examinations.