Combinations
\(C(5,2) = 10\) - Choose 2 from 5: {AB, AC, AD, AE, BC, BD, BE, CD, CE, DE}
Combinatorics Calculator
Calculate combinations C(n,r), permutations P(n,r), and factorials n! with detailed explanations and step-by-step solutions
Results
Enter values and click Calculate to see the result.
Combinatorics
Combinatorics is the mathematics of counting. It helps answer questions like "How many ways?" and "How many possibilities?"
Formulas:
- Factorial: \(n! = n \times (n-1) \times \cdots \times 2 \times 1\), with \(0! = 1\)
- Permutations: \(P(n,r) = \frac{n!}{(n-r)!}\) - arrangements where order matters
- Combinations: \(C(n,r) = \frac{n!}{r! \times (n-r)!}\) - selections where order doesn't matter
Key Differences:
- Combinations: Use when order doesn't matter (teams, groups, selections)
- Permutations: Use when order matters (rankings, arrangements, sequences)
- Factorial: Use for total arrangements of all items
Applications:
- Probability calculations
- Password and code possibilities
- Tournament brackets and schedules
- Card games and lottery odds
- Genetics and DNA sequences
Worked Examples
Permutations
\(P(5,2) = 20\) - Arrange 2 from 5: AB, BA, AC, CA, AD, DA, ... (order matters)
Factorial
\(5! = 120\) - Ways to arrange 5 items: \(5 \times 4 \times 3 \times 2 \times 1\)