Permutation & Combination
Factorial
- Definition: n! = n × (n-1) × (n-2) × ... × 2 × 1
- Example: 5! = 5 × 4 × 3 × 2 × 1 = 120
- 0! = 1 (by definition)
- 1! = 1
Permutation (Order matters)
- Definition: Arrangement of objects where order is important
- Formula: ⁿPᵣ = n!/(n-r)!
- All arrangements of n: n!
- Example: ABC, ACB, BAC, BCA, CAB, CBA (6 permutations)
Combination (Order doesn't matter)
- Definition: Selection of objects where order is not important
- Formula: ⁿCᵣ = n!/[r!(n-r)!]
- Example: AB, AC, BC (3 combinations from A,B,C)
Important Properties
- ⁿCᵣ = ⁿCₙ₋ᵣ
- ⁿC₀ = ⁿCₙ = 1
- ⁿC₁ = n
- ⁿPᵣ = r! × ⁿCᵣ
- ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ
Special Cases
- Circular arrangement: (n-1)!
- Arrangement with repetition: n!/p!q!r! (p, q, r identical items)
- Selection with repetition: ⁿ⁺ʳ⁻¹Cᵣ
When to Use What?
- Use Permutation: When order matters (arrangement, ranking, passwords)
- Use Combination: When order doesn't matter (selection, committee, groups)
Quick Tips
- If question uses "arrange" or "order" → Permutation
- If question uses "select" or "choose" → Combination
- Permutation > Combination (for same n and r)
- Memorize factorial values up to 10!