Last Updated: June 1, 2026

📋 Table of Contents
What is Permutations Combinations Revision Notes?
Introduction
1. Fundamental Principle of Counting (FPC)
2. Factorials (n!)
3. Permutations (Arrangements)
4. Combinations (Selections)
5. Difference: Permutation vs Combination
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics
📚 Related Topics

📋 Table of Contents

What is Permutations Combinations Revision Notes?
Introduction
1. Fundamental Principle of Counting (FPC)
2. Factorials (n!)
3. Permutations (Arrangements)
4. Combinations (Selections)
5. Difference: Permutation vs Combination
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics

Permutations Combinations Class 11 Mathematics Revision — JEE 2026 Grandmaster Guide

What is Permutations Combinations Revision Notes?

[!TIP] 🚀 2-Minute Quick Recall Summary (Save for Exam Day)

Fundamental Principle of Counting (FPC):

And (Multiplication): If task A and m ways AND task B and and ways -> m × n ways.

Or (Addition): If task A and m ways OR task B and and ways -> m + n ways.

Factorial (n!): n! = n × (n-1) × ... × 1. (0! = 1).

Permutation (nPr): Arrangement where order MATTERS. nPr = n! / (n - r)!.

Combination (nCr): Selection where order DOES NOT matter. nCr = n! / [r!(n - r)!].

Relation: nPr = nCr × r!. 📥 Download 1-Page Short Notes PDF (Zero-Friction)

Introduction

Permutations and Combinations provide the mathematical toolkit for counting possibilities and arrangements and complex systems. Master the Fundamental Principle of Counting, nPr for ordered arrangements, n nCr for selections to excel and probability and cryptography. This class 11 Math Chapter 7 guide covers all essential factorial logic for JEE and CBSE exams. Mathematics is not just about numbers; it's about possibilities.

1. Fundamental Principle of Counting (FPC)

This is the base of all counting techniques.

Multiplication Principle: If an event occurs n m different ways, following which another event occurs n n different ways, then the total number of occurrence of the events and the given order is m × n.
Addition Principle: If an event can occur n m ways and another independent event can occur n n ways, then either of the two events can occur n m + n ways.

2. Factorials (n!)

The product of first n natural numbers is called n-factorial.

n! = 1 × 2 × 3 × ... × n.
0! = 1 (y definition).
n! = n × (n - 1)!.

3. Permutations (Arrangements)

A permutation is an arrangement and a definite order of a number of objects taken some or all at a time.

theorem 1: The number of permutations of and different objects taken r at a time (0 < r ≤ n) n objects do not repeat is nPr = n! / (n - r)!.
theorem 2: If repetition is allowed, the number of permutations is nʳ.
theorem 3: If out of and objects, p are of one kind, q of another, n the rest are different, number of permutations = n! / (p!q!).

4. Combinations (Selections)

A combination is a selection of items where the order of selection does not matter.

Theorem: The number of combinations of and different objects taken r at a time is nCr = n! / [r!(n - r)!].
Properties:
1. nCr = nC(n-r)
2. nCa = nCb => either a = b or a + b = n.
3. nCr + nC(r-1) = (n+1)Cr (Pascal's Formula).

5. Difference: Permutation vs Combination

Feature	Permutation	Combination
Focus	Arrangement / Order	Selection / Grouping
Order	Matters	Does not matter
Keyword	Arrange, List, Align	Select, Choose, Pick
Formula	nPr	nCr

Comprehensive Exam Strategy (Q&A)

Q1: How many 3-digit numbers can be formed from the digits 1, 2, 3, 4, 5 assuming that repetition of digits is allowed? Answer:

Total digits = 5. Places to fill = 3.
Using FPC: 5 × 5 × 5 = 125 ways.

Q2: Find and if n-1P3 : nP4 = 1 : 9. Answer:

[(n-1)! / (n-1-3)!] / [n! / (n-4)!] = 1/9
[(n-1)! / (n-4)!] × [(n-4)! / n!] = 1/9
(n-1)! / n! = 1/9
(n-1)! / n(n-1)! = 1/9
n = 9.

Q3: A committee of 3 persons is to be constituted from a group of 2 men 3 women. In how many ways can this be done? Answer:

Order doesn't matter, so use combinations.
Total people = 2 + 3 = 5. Select 3.
5C3 = 5! / (3!2!) = (5 × 4) / 2 = 10 ways.

Related Revision Notes

Chapter 6: linear Inequalities
Chapter 8: Binomial Theorem
[External Reference: NCERT Class 11 Math Chapter 7 (Authoritative Source)]

Conclusion

permutations n Combinations transform the way we see complexity. By mastering the core formulas of nPr and nCr, n understanding when order matters, you gain the power to calculate outcomes and everything from poker hands to the number of ways to sequence DNA. Keep your factorials small and your logic sharp!

This post was curated by Jules, Exam Compass Bot, and edited for accuracy y Ayush.

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Continue your revision with these related guides:

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

🎬 Watch video explanations on YouTube →

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🪤 The 5 Mistakes That Cost Marks

Forgetting to Apply the Formula for Permutations with Repetition: When items can be repeated, the formula for permutations is n^r, where and is the number of items to choose from and r is the number of items being chosen. Many students forget to apply this formula, leading to incorrect answers.
Confusing Permutations and Combinations: Permutations consider the order of selection, while combinations do not. A common mistake is to use the combination formula when the problem requires permutations, and vice versa.
Not Considering the Case of Non-Distinct Objects: When dealing with permutations of non-distinct objects, the formula for permutations needs to be adjusted. Many students fail to account for this, resulting in incorrect calculations.
Incorrectly Applying the Formula for Combinations with Repetition: The formula for combinations with repetition is (n+r-1) choose r, where and is the number of items to choose from and r is the number of items being chosen. A common mistake is to misapply this formula, leading to incorrect answers.
Failing to Account for Overcounting: In certain problems, overcounting can occur when the same arrangement is counted multiple times. Students often forget to adjust for overcounting, resulting in an incorrect final answer.

🔁 Last 5 Minutes Box

Permutations: nPr = n! / (n-r)!,* Combinations: nCr = n! / (r!(n-r)!),* Important Identities: nCr + nC(r-1) = (n+1)Cr, nCr = nC(n-r),* Properties: nC0 = nCn = 1, nC1 = n, nCn = 1,* Applications: Dividing objects into groups, Selecting items from a set

Last Updated: June 1, 2026

📋 Table of Contents
What is Permutations Combinations Revision Notes?
Introduction
1. Fundamental Principle of Counting (FPC)
2. Factorials (n!)
3. Permutations (Arrangements)
4. Combinations (Selections)
5. Difference: Permutation vs Combination
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics
📚 Related Topics

📋 Table of Contents

What is Permutations Combinations Revision Notes?
Introduction
1. Fundamental Principle of Counting (FPC)
2. Factorials (n!)
3. Permutations (Arrangements)
4. Combinations (Selections)
5. Difference: Permutation vs Combination
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics

Permutations Combinations Class 11 Mathematics Revision — JEE 2026 Grandmaster Guide

What is Permutations Combinations Revision Notes?

[!TIP] 🚀 2-Minute Quick Recall Summary (Save for Exam Day)

Fundamental Principle of Counting (FPC):

And (Multiplication): If task A and m ways AND task B and and ways -> m × n ways.

Or (Addition): If task A and m ways OR task B and and ways -> m + n ways.

Factorial (n!): n! = n × (n-1) × ... × 1. (0! = 1).

Permutation (nPr): Arrangement where order MATTERS. nPr = n! / (n - r)!.

Combination (nCr): Selection where order DOES NOT matter. nCr = n! / [r!(n - r)!].

Relation: nPr = nCr × r!. 📥 Download 1-Page Short Notes PDF (Zero-Friction)

Introduction

1. Fundamental Principle of Counting (FPC)

This is the base of all counting techniques.

Multiplication Principle: If an event occurs n m different ways, following which another event occurs n n different ways, then the total number of occurrence of the events and the given order is m × n.
Addition Principle: If an event can occur n m ways and another independent event can occur n n ways, then either of the two events can occur n m + n ways.

2. Factorials (n!)

The product of first n natural numbers is called n-factorial.

n! = 1 × 2 × 3 × ... × n.
0! = 1 (y definition).
n! = n × (n - 1)!.

3. Permutations (Arrangements)

A permutation is an arrangement and a definite order of a number of objects taken some or all at a time.

theorem 1: The number of permutations of and different objects taken r at a time (0 < r ≤ n) n objects do not repeat is nPr = n! / (n - r)!.
theorem 2: If repetition is allowed, the number of permutations is nʳ.
theorem 3: If out of and objects, p are of one kind, q of another, n the rest are different, number of permutations = n! / (p!q!).

4. Combinations (Selections)

A combination is a selection of items where the order of selection does not matter.

Theorem: The number of combinations of and different objects taken r at a time is nCr = n! / [r!(n - r)!].
Properties:
1. nCr = nC(n-r)
2. nCa = nCb => either a = b or a + b = n.
3. nCr + nC(r-1) = (n+1)Cr (Pascal's Formula).

5. Difference: Permutation vs Combination

Feature	Permutation	Combination
Focus	Arrangement / Order	Selection / Grouping
Order	Matters	Does not matter
Keyword	Arrange, List, Align	Select, Choose, Pick
Formula	nPr	nCr

Comprehensive Exam Strategy (Q&A)

Q1: How many 3-digit numbers can be formed from the digits 1, 2, 3, 4, 5 assuming that repetition of digits is allowed? Answer:

Total digits = 5. Places to fill = 3.
Using FPC: 5 × 5 × 5 = 125 ways.

Q2: Find and if n-1P3 : nP4 = 1 : 9. Answer:

[(n-1)! / (n-1-3)!] / [n! / (n-4)!] = 1/9
[(n-1)! / (n-4)!] × [(n-4)! / n!] = 1/9
(n-1)! / n! = 1/9
(n-1)! / n(n-1)! = 1/9
n = 9.

Q3: A committee of 3 persons is to be constituted from a group of 2 men 3 women. In how many ways can this be done? Answer:

Order doesn't matter, so use combinations.
Total people = 2 + 3 = 5. Select 3.
5C3 = 5! / (3!2!) = (5 × 4) / 2 = 10 ways.

Related Revision Notes

Chapter 6: linear Inequalities
Chapter 8: Binomial Theorem
[External Reference: NCERT Class 11 Math Chapter 7 (Authoritative Source)]

Conclusion

This post was curated by Jules, Exam Compass Bot, and edited for accuracy y Ayush.

📚 Related Topics

Continue your revision with these related guides:

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

🎬 Watch video explanations on YouTube →

📚 Related Topics

Continue your revision with these related guides:

🪤 The 5 Mistakes That Cost Marks

Forgetting to Apply the Formula for Permutations with Repetition: When items can be repeated, the formula for permutations is n^r, where and is the number of items to choose from and r is the number of items being chosen. Many students forget to apply this formula, leading to incorrect answers.
Confusing Permutations and Combinations: Permutations consider the order of selection, while combinations do not. A common mistake is to use the combination formula when the problem requires permutations, and vice versa.
Not Considering the Case of Non-Distinct Objects: When dealing with permutations of non-distinct objects, the formula for permutations needs to be adjusted. Many students fail to account for this, resulting in incorrect calculations.
Incorrectly Applying the Formula for Combinations with Repetition: The formula for combinations with repetition is (n+r-1) choose r, where and is the number of items to choose from and r is the number of items being chosen. A common mistake is to misapply this formula, leading to incorrect answers.
Failing to Account for Overcounting: In certain problems, overcounting can occur when the same arrangement is counted multiple times. Students often forget to adjust for overcounting, resulting in an incorrect final answer.

🔁 Last 5 Minutes Box

Permutations: nPr = n! / (n-r)!,* Combinations: nCr = n! / (r!(n-r)!),* Important Identities: nCr + nC(r-1) = (n+1)Cr, nCr = nC(n-r),* Properties: nC0 = nCn = 1, nC1 = n, nCn = 1,* Applications: Dividing objects into groups, Selecting items from a set