Online Combinations & Permutations Calculator
Solve combinatorics problems with combinations nCr, permutations nPr, and factorial operations. Choose between sampling with or without replacement.
Combinations & Permutations Calculator
Calculate combinations, permutations, and variations with or without replacement, step-by-step math and list generator
Real-world presets:
Formula:
C(10, 3) = 10! / (3! × (10 - 3)!)Step-by-step solution:
10! / (3! × 7!) = 120
120
Quick Guide: Which formula to use?
A(n, k) = n! / (n - k)!
Order matters, choosing subset k of n → Arrangements A(n, k) / nPr
P(n) = n!
Order matters, using ALL n elements → Permutations P(n) = n!
C(n, k) = n! / (k!(n - k)!)
Order does NOT matter (groups, teams, hands) → Combinations C(n, k) / nCr
How to use
- Enter total number of items n.
- Enter sample subset size k.
- Choose repetition option (with or without replacement).
- View computed combination count, permutation count, and formula breakdown.
Combinatorial Formulas
Permutations without repetition: P(n) = n!
Permutations of subset: nPr = n! / (n - k)!
Combinations without repetition: nCr = n! / (k! * (n - k)!)
Combinations with repetition: (n + k - 1)! / (k! * (n - 1)!)
Calculation Methodology
Permutations account for ordering, whereas combinations treat different permutations of the same subset as equivalent.
Frequently Asked Questions
What is the distinction between combinations and permutations?
Order matters for permutations (like passcodes), whereas order does not matter for combinations (like a hand of cards).
What is a factorial n!?
The product of all positive integers less than or equal to n, with 0! defined as 1.
Can n and k be negative?
No, standard combinatorial sampling requires non-negative integers where k ≤ n without replacement.
How are large factorials computed without overflow?
Fractions are simplified by canceling shared factorial terms prior to multiplication.
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