Last Updated: April 2026
Permutations and Combinations is one of the most scoring chapters in JEE Main Mathematics, with 2–3 questions guaranteed every year. In JEE Main 2025 (Session 1), there were 3 questions from P&C — contributing 12 marks directly. The key advantage: this chapter has fewer formulas than calculus but requires sharp logical application. Master this guide for JEE Main 2027.
Fundamental Counting Principle
If event A can occur in m ways and event B in n ways independently, then:
- Multiplication Rule (AND): A AND B = m × n ways
- Addition Rule (OR): A OR B = m + n ways (when mutually exclusive)
Factorials
n! = n × (n−1) × (n−2) × … × 2 × 1
Special cases: 0! = 1, 1! = 1
| n | n! |
|---|---|
| 1 | 1 |
| 2 | 2 |
| 3 | 6 |
| 4 | 24 |
| 5 | 120 |
| 6 | 720 |
| 7 | 5040 |
| 10 | 3,628,800 |
Permutations (Arrangement)
An arrangement where order matters.
Key Formulas
- ⁿPᵣ = n! / (n−r)! — arrangements of r objects from n distinct objects
- n! — arrangements of all n distinct objects in a row
- Circular permutation: (n−1)! for n distinct objects in a circle
- Necklace/garland: (n−1)!/2 (clockwise = anticlockwise)
Permutations with Repetitions
- n objects with p identical, q identical, r identical: n! / (p! × q! × r!)
- r objects from n with unlimited repetition allowed: nʳ
Important Results
| Scenario | Formula | Example |
|---|---|---|
| Arrange n distinct in a line | n! | 5 books: 5! = 120 |
| r of n distinct in a line | ⁿPᵣ | Select 3 from 8 and arrange: ⁸P₃ = 336 |
| n in circle | (n−1)! | 8 around table: 7! = 5040 |
| MISSISSIPPI | 11!/(4!4!2!) | = 34,650 |
| Specific objects always together | Treat as 1 unit, multiply by internal arrangements | A & B always together: (n−1)! × 2! |
| Specific objects never together | Total − (arrangements with them together) | Complementary approach |
Combinations (Selection)
A selection where order does NOT matter.
Key Formulas
- ⁿCᵣ = n! / [r! × (n−r)!]
- ⁿCᵣ = ⁿCₙ₋ᵣ (symmetry property)
- ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ (Pascal’s identity)
- ⁿC₀ = ⁿCₙ = 1
Key Values to Memorize
| Expression | Value |
|---|---|
| ¹⁰C₃ | 120 |
| ¹⁰C₄ | 210 |
| ¹²C₅ | 792 |
| ⁸C₃ | 56 |
| ⁶C₃ | 20 |
| ²⁰C₂ | 190 |
Applications — Geometry
Points, Lines, Triangles, Diagonals
- Lines through n points (no 3 collinear): ⁿC₂
- Triangles from n points (no 3 collinear): ⁿC₃
- Diagonals of n-sided polygon: ⁿC₂ − n = n(n−3)/2
- Regions in a circle (n chords, no 3 concurrent): ⁿC₂ + ⁿC₀ + 1
Example — Diagonals of a Hexagon
n = 6: Diagonals = 6(6−3)/2 = 6×3/2 = 9
Distribution Problems
| Type | Formula |
|---|---|
| n distinct objects into r distinct groups (no restriction) | rⁿ |
| n identical objects into r distinct groups (no zero) | ⁿ⁻¹Cᵣ₋₁ |
| n identical objects into r distinct groups (zeros allowed) | ⁿ⁺ʳ⁻¹Cᵣ₋₁ |
| n distinct objects into r identical groups | Stirling numbers (advanced) |
Derangements
Number of ways to arrange n objects so that NO object is in its original position:
D(n) = n! × [1 − 1/1! + 1/2! − 1/3! + … + (−1)ⁿ/n!]
- D(2) = 1, D(3) = 2, D(4) = 9, D(5) = 44
Binomial Coefficient Connection
- Sum of all combinations: ⁿC₀ + ⁿC₁ + … + ⁿCₙ = 2ⁿ
- Sum of even-indexed: ⁿC₀ + ⁿC₂ + … = 2ⁿ⁻¹
- Sum of odd-indexed: ⁿC₁ + ⁿC₃ + … = 2ⁿ⁻¹
JEE Main Strategy — P&C
- Read carefully — distinguish “arrangement” (P) from “selection” (C)
- Identify restrictions — objects that must be together, apart, or at specific positions
- Complementary counting — sometimes easier to count what you DON’T want and subtract from total
- Case-by-case — when restrictions split the problem, enumerate cases
- Verify with small n — if formula is uncertain, test with n=2 or n=3 manually
Practice MCQs — JEE Main P&C 2027
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Frequently Asked Questions (FAQ)
What is the difference between permutation and combination?
Permutation is arrangement where order matters (e.g., choosing a president, VP, secretary from a group). Combination is selection where order does NOT matter (e.g., choosing a committee). The formula is ⁿPᵣ = n!/(n−r)! for permutations and ⁿCᵣ = n!/[r!(n−r)!] for combinations. Always ⁿPᵣ ≥ ⁿCᵣ, since ⁿPᵣ = ⁿCᵣ × r!.
How many questions come from Permutations and Combinations in JEE Main?
JEE Main consistently has 2–3 questions from Permutations and Combinations. The chapter is part of “Algebra” in the JEE Main syllabus. Questions range from direct formula application to complex application problems involving geometry (diagonals, triangles) or number theory.
What is a derangement?
A derangement is a permutation where no element appears in its original position. For example, if 3 letters are to be placed in 3 envelopes with none in the correct envelope, the number of derangements = D(3) = 2. The general formula is D(n) = n! × Σ (−1)^k/k! for k from 0 to n.
How many diagonals does a regular polygon with n sides have?
A polygon with n sides has n(n−3)/2 diagonals. This comes from ⁿC₂ (total lines joining any two vertices) minus n (the sides themselves). For example, a hexagon (n=6) has 6×3/2 = 9 diagonals; a decagon (n=10) has 10×7/2 = 35 diagonals.
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