Combinatória / Probabilidade — Calculadora Online Grátis
nPr and nCr calculations with factorial steps and formula explanation
The Permutation & Combination Calculator computes P(n,r) and C(n,r) simultaneously, showing the factorial expansion at each step. Permutations count ordered arrangements; combinations count unordered selections. Probability students, exam takers, and game designers use this tool to count possibilities without manually computing large factorials.
Frequently Asked Questions
Sobre Esta Calculadora
Counting principles are the foundation of probability theory. Permutations (P(n,r) = n!/(n−r)!) count the number of ways to arrange r items from a set of n where order matters. Combinations (C(n,r) = n!/(r!×(n−r)!)) count the number of ways to choose r items where order does not matter. The distinction is critical: choosing a 3-member team from 8 people gives C(8,3) = 56 ways, but assigning them first, second, and third place gives P(8,3) = 336 ways.
This calculator shows the full factorial expansion, making it clear how the denominator cancels terms. For large n and small r, computing P or C directly via factorials can overflow — the calculator handles this gracefully.
Applications include probability problems (lottery odds, card game hands), combinatorics in algorithm analysis (Big-O notation), DNA sequence counting in bioinformatics, and scheduling problems in operations research.
Efetue sem dor a probabilidade das matrizes de combinações em loterias.
Fórmula
P(n,r) = n! / (n-r)! C(n,r) = n! / (r! × (n-r)!)
Exemplo Resolvido
How many ways can you choose 3 items from 8 (ordered vs. unordered)?
Entradas:
- n = 8 (total items)
- r = 3 (items chosen)
Passo a Passo:
- Permutation P(8,3) — order matters:
- P(8,3) = 8! / (8−3)! = 8! / 5!
- = (8 × 7 × 6 × 5!) / 5! = 8 × 7 × 6 = 336
- Combination C(8,3) — order does not matter:
- C(8,3) = 8! / (3! × 5!) = 336 / 3! = 336 / 6 = 56