Combinations
\(C(5,2) = 10\) - Choose 2 from 5: {AB, AC, AD, AE, BC, BD, BE, CD, CE, DE}
Kombinatorik-Rechner
Berechnen Sie Kombinationen C(n,r), Permutationen P(n,r) und Fakultäten n! mit detaillierten Erklärungen und Schritt-für-Schritt-Lösungen
Ergebnisse
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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: \(5 \times 4 \times 3 \times 2 \times 1\)