Math & CombinatoricsVerified: 2026-10-05
Permutation and Combination Calculator
Calculate permutations (nPr - order matters) and combinations (nCr - order does not matter) with and without repetition, factorials, and step-by-step arithmetic.
Direct Answer:Permutations (nPr = n! / (n - r)!) count arrangements where order matters (e.g. 10P3 = 720). Combinations (nCr = n! / [r!(n - r)!]) count selections where order does not matter (e.g. 10C3 = 120).
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Adjust Parameters
0100
Total elements in the set (n ≥ r)
0100
Number of elements to arrange or group
Enable if items can be chosen multiple times (e.g. dice rolls or passcode digits).
Combinations: 10C3
120
Permutations (10P3): 720
Choosing 3 from 10 gives 120 unique groups (order ignored) and 720 ordered sequences.
Permutations (10P3 - Order Matters)
720
Combinations (10C3 - Order Doesn't Matter)
120
Repetition Allowed?
No (Without Replacement)
Step-by-Step Combinatorics Derivation
- Given total items n = 10, sample size r = 3:
- 1. Permutations (Order Matters) Formula: nPr = n! / (n − r)!
- 10P3 = 10! / (10 − 3)! = 10! / 7! = 720
- 2. Combinations (Order Does NOT Matter) Formula: nCr = n! / [r! × (n − r)!]
- 10C3 = 10! / [3! × 7!] = 120
- 3. Relation: nCr = nPr / r! = 720 / 3! = 120
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How Permutation and Combination Calculator is Calculated
Transparent mathematical formulas and assumptions used across our financial engines.
Mathematical Formula
nPr = \frac{n!}{(n - r)!} \quad | \quad nCr = \binom{n}{r} = \frac{n!}{r!(n - r)!} \quad | \quad nCr = \frac{nPr}{r!}
Variables & Legend
n
Total Elements
r
Sample Size
nPr
Permutations
nCr
Combinations
Step-by-Step Calculation
- Factorial Expansion:
n! = n \times (n-1) \times \dots \times 1Compute factorials for n, r, and (n - r).
- Permutations:
nPr = n! / (n - r)!Divide n! by (n - r)! to find ordered permutations.
- Combinations:
nCr = nPr / r!Divide permutations by r! to remove order significance.
Conventions & Assumptions
- Permutations are always greater than or equal to combinations (nPr ≥ nCr).
- 0! is defined mathematically as 1 (0! = 1).
- When r = 0, both nP0 = 1 and nC0 = 1.
Frequently Asked Questions
Order. In permutations, order matters (e.g., a locker combination 1-2-3 is different from 3-2-1). In combinations, order does not matter (e.g., choosing a team of Alice, Bob, and Charlie is the same as Charlie, Alice, and Bob).