Permutation/Combination Calculator
Calculate permutations (P) and combinations (C) with step-by-step solutions
P(n, r) = ?
Select r from n items considering order
Permutation
P(n, r) = n! / (n - r)!
When order matters
e.g., ABC, ACB, BAC are all different
Combination
C(n, r) = n! / (r! × (n - r)!)
When order doesn't matter
e.g., ABC, ACB, BAC are all the same
Real-life Examples
Permutation Examples
- • Creating passwords
- • Determining 1st, 2nd, 3rd place in race
- • Seat arrangements
Combination Examples
- • Team formation
- • Menu selection
- • Committee formation
Historical Development of Combinatorics
Combinatorics began in ancient India and China and has evolved into a core field of modern mathematics. From Pascal's triangle and the binomial theorem to modern graph theory and cryptography, combinatorics has played a crucial role in the development of mathematics and computer science.
Ancient and Medieval
- • India (2nd century BC): Pingala's binomial coefficients
- • China (11th century): Yang Hui's triangle
- • Islamic world (12th century): Al-Karaji's combinatorics
- • Europe (13th century): Fibonacci's combinatorial problems
Modern Era
- • Pascal (17th century): Connecting probability and combinatorics
- • Euler (18th century): Generating function theory
- • 20th century: Graph theory, design theory
- • Modern: Integration with computer science
Connection to Probability Theory
Basic Probability Calculations
Classical probability: P(A) = favorable outcomes / total outcomes
Permutation probability: probability of order-dependent events
Combination probability: probability of order-independent events
Conditional probability: probability under specific conditions
Practical Application Examples
Card games: calculating poker hand probabilities
Lottery: calculating winning probabilities
Quality control: sample inspection probabilities
Genetics: gene combination probabilities
🎯 Combinatorics Mastery Strategy
• Problem classification: First determine importance of order to distinguish permutation/combination.
• Step-by-step approach: Break complex problems into smaller units.
• Use symmetry: Find problem symmetries to simplify calculations.
• Practical application: Apply combinatorics to daily problems to build intuition.
• Programming practice: Implement combinatorial algorithms through coding.