Permutation & Combination Calculator
Calculate permutations P(n,r) and combinations C(n,r) with detailed step-by-step solutions and factorial calculations.
Permutation & Combination Calculator
P(5,3) = 5! / (5-3)!
About Permutations & Combinations
Permutations and combinations are fundamental concepts in combinatorics that deal with counting arrangements and selections of objects.
Key Differences
- Permutation: Order matters (arrangements)
- Combination: Order doesn't matter (selections)
- P(n,r) ≥ C(n,r): Permutations are always greater than or equal to combinations
Formulas
- Permutation: P(n,r) = n! / (n-r)!
- Combination: C(n,r) = n! / (r! × (n-r)!)
- Relationship: P(n,r) = r! × C(n,r)
Examples
- Permutation: Arranging 3 people in a line from 5 people = P(5,3) = 60
- Combination: Choosing 3 people from 5 people = C(5,3) = 10
- Password creation: Use permutations (order matters)
- Team selection: Use combinations (order doesn't matter)
Special Cases
- P(n,0) = C(n,0) = 1 (choosing nothing)
- P(n,n) = n! (all arrangements)
- C(n,n) = 1 (choosing everything)
- C(n,r) = C(n,n-r) (symmetry property)
Applications
- Probability: Calculating favorable outcomes
- Statistics: Sampling and experimental design
- Computer Science: Algorithm analysis, cryptography
- Game Theory: Strategy analysis
- Genetics: Gene combination analysis
Real-World Examples
- Lottery: C(49,6) for choosing 6 numbers from 49
- Seating arrangements: P(8,8) for 8 people at a round table
- Committee formation: C(20,5) for selecting 5 members from 20
- Password security: P(26,8) for 8-character passwords