Chapter 22 · Class 11-12 & JEE Main/Adv
Permutations & Combinations
56 topics in 16 parts
Permutations Combinations Factorial Zero Factorial Fundamental Principle of Counting Addition Principle of Counting
Key ideas
Analytical Comparison & Distinction Matrix
| Analytical Method | Standard Representation | Parametric / Invariant Framework |
|---|---|---|
| Algebraic Nature | Explicit equations relating spatial coordinates | Parameterized using scalar parameters (λ, μ) or vectors |
| Symmetry Usage | Relies on coordinate expansion | Exploits vector dot/cross products for rapid simplification |
| Computational Efficiency | Direct for intercept evaluation | Superior for angles, orthogonal projections, and distances |
| Common Mistake | Sign confusion in determinants | Omitting boundary condition checks |
All 56 topics in Permutations & Combinations
Fundamental Principle of Counting
- Addition Rule
- Multiplication Rule
- Complementary & Symmetry Methods
Factorial Notation
- Prime Exponent in n!
- Identities & Sums
- Trailing & Last Non-Zero Digits
Permutation
- Distinct Objects
- Repetition Allowed
- Identical Objects
- Restricted Arrangements
- Word Arrangements
- Lexicographic Ordering
Combination
- Basic Selection & Properties
- Restricted Selections
- Committee & Team Formation
- Subset Counting
- Selection in Arithmetic & Geometric Progressions
Circular Permutations
- Clockwise & Anticlockwise Arrangements
- Restricted Arrangements
- Arrangements with Identical Objects
All Possible Selections
- Selections from Distinct Objects
- Selections from Identical Objects
- Mixed Selections
Divisions & Distributions
- Distribution of Distinct Objects
- Division into Groups
- Balls & Boxes
Distribution of Identical Objects
- Stars & Bars
- Positive Sum Solutions
- Non-Negative Sum Solutions
Multinomial Theorem
- Coefficients & Expansion Terms
- Counting Applications
Principles of Inclusion & Exclusion
- Set-Theoretic Applications
- Surjective Mappings
Permutations & Combinations in Number Theory
- Number of Divisors
- Ordered Factorizations
- Divisibility-Based Counting
- Residue-Class Counting
Derangement
- Complete Derangements
- Partial Derangements
Geometric Counting
- Lines, Triangles & Polygons
- Regions & Intersections
- Grid & Lattice Paths
- Chessboard Counting
Counting Functions
- Total Functions
- Injective Functions
- Surjective Functions
- Monotone Functions
- Restricted Mappings
Number Formation & Digit Counting
- Digit-Sum Constraints
- Divisibility Constraints
- Repeated-Digit Constraints
- Leading-Zero Restrictions
- General Number Formation
Combinatorial Identities
- Properties of nCr
- Permutation Identities
- Recurrences & Series Sums