The combination calculator counts how many ways there are to choose r items from n. Choose 6 numbers from 59, as in UK Lotto, and there are 45,057,474 combinations: that’s why one line has a 1 in 45 million chance in each Lotto round. It’s free, and it also handles permutations, where the order matters, and choices where repeats are allowed.
Combination calculator: which type do you need?
| Type | Order matters? | Repeats? | Example | Ways |
|---|---|---|---|---|
| Combinations (nCr) | No | No | 5-card poker hand from 52 | 2,598,960 |
| Combinations with repeats | No | Yes | 3 scoops from 5 flavours | 35 |
| Permutations (nPr) | Yes | No | 1st, 2nd and 3rd from 10 runners | 720 |
| Permutations with repeats | Yes | Yes | 4-digit PIN | 10,000 |
The test is simple: if swapping two items gives you something different, order matters. A lottery line 4-8-15-16-23-42 is the same ticket in any order, so it’s a combination. A tricast 4-8-15 is a different bet from 15-8-4, so it’s a permutation.
The formulas
- Combinations: nCr = n! ÷ (r! × (n − r)!)
- Permutations: nPr = n! ÷ (n − r)!
- Combinations with repeats: (n + r − 1)! ÷ (r! × (n − 1)!)
- Permutations with repeats: nʳ
Permutations are always the bigger number, by a factor of r!. 10 runners give 120 ways to pick three of them, but 720 ways to pick them in order, because each group of three can finish 6 ways.
Combinations in gambling
Lottery odds are combinations. EuroMillions is 5 from 50 times 2 Lucky Stars from 12: 2,118,760 × 66 = 139,838,160 possible tickets. Poker probabilities start here too: of the 2,598,960 five-card hands, just 4 are royal flushes. And betting systems are combinations of selections: 4 picks make 6 doubles (4C2), 4 trebles (4C3) and 1 fourfold, which is a Yankee.
For full lottery prize odds, use the lottery odds calculator. For a chance of something happening at least once, the probability calculator is the next step.
Combinations are the building block of card-draw odds. The hypergeometric calculator uses them to give the chance of drawing a number of hits from a deck, with versions for MTG and Yu-Gi-Oh.