Combinations Calculator (nCr)
How many ways to choose r items from n when order does not matter — nCr, with the permutation count alongside for comparison.
A combination counts selections, not arrangements: choosing Alice then Bob is the same committee as Bob then Alice.
How the combinations calculator (ncr) works
A combination counts selections, not arrangements: choosing Alice then Bob is the same committee as Bob then Alice. So take the permutation count nPr = n! ÷ (n − r)! and divide out the r! orderings of each selection.
The result is symmetric: choosing 3 from 10 has the same count as choosing 7 from 10, because every choice of 3 leaves a choice of 7.
Formula: C(n, r) = n! / (r! (n − r)!)
Worked examples
| Inputs | Combinations C(n, r) | Note |
|---|---|---|
| Choose 3 from 10 | 120 | 120 |
| Lotto: 6 from 45 | 8,145,060 | 8,145,060 |
| A 5-card poker hand from 52 | 2,598,960 | 2,598,960 |
FAQFrequently asked questions
What is the difference between a combination and a permutation?
A permutation counts arrangements, a combination counts selections. Order matters in the first and not the second.
How many 6-from-45 lotto tickets are there?
8,145,060 — so a single ticket has about a 1 in 8.1 million chance of the jackpot.
Why does C(10,3) equal C(10,7)?
Because choosing 3 to include is the same act as choosing 7 to leave out.
What is 0 choose 0?
One. There is exactly one way to choose nothing from nothing — the empty selection.
How many poker hands are there?
C(52, 5) = 2,598,960 five-card hands.
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)
- Common Core State Standards — Mathematics — the terms and methods taught in US schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.