Combinations
\(C(5,2) = 10\) - Choose 2 from 5: {AB, AC, AD, AE, BC, BD, BE, CD, CE, DE}
Calculatrice de combinatoire
Calculer les combinaisons C(n,r), permutations P(n,r) et factorielles n! avec explications détaillées et solutions étape par étape
Résultats
Entrez les valeurs et cliquez sur Calculer pour voir le résultat.
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: \(5 \times 4 \times 3 \times 2 \times 1\)