Permutations and Combinations Calculator
How many ways can you choose r items from n — when order matters (permutations, nPr) and when it doesn't (combinations, nCr)?
Permutations count ordered selections: nPr = n! ÷ (n − r)!.
How the permutations and combinations calculator works
Permutations count ordered selections: nPr = n! ÷ (n − r)!. Combinations count unordered ones, so divide out the r! orderings: nCr = n! ÷ (r!(n − r)!). Picking a lottery line is a combination; a race podium is a permutation.
Formula: nPr = n! / (n − r)!; nCr = n! / (r! (n − r)!)
Worked examples
| Inputs | Combinations (nCr) | Note |
|---|---|---|
| 6 from 45 (lottery) | 8145060 | 8,145,060 combinations |
| 3 from 10 podium | 120 | 720 permutations |
| 2 from 5 | 10 | 10 pairs, 20 ordered pairs |
FAQFrequently asked questions
What is the difference between a permutation and a combination?
Permutations count order (AB ≠ BA); combinations do not (AB = BA). There are always r! times more permutations.
What are the odds of winning a 6-from-45 lottery?
1 in 8,145,060 for a single line.
What if repetition is allowed?
Ordered selections with repetition are nʳ — a 4-digit PIN has 10⁴ = 10,000 possibilities.
What is nC0 or nCn?
1 — there is one way to choose nothing and one way to choose everything.
Why does the calculator stop at n = 1000?
The numbers become astronomically large; beyond that use logarithms.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.