🎲 Permutation & Combination
Enter n and r to count ordered and unordered selections, with and without repeats, using exact integer arithmetic so large results are not rounded.
Quick answer: nPr = n! ÷ (n − r)! and nCr = n! ÷ (r! × (n − r)!). For n = 10 and r = 3, nPr = 720 and nCr = 120.
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Permutation & Combination inputs
Result
Combinations C(10, 3)
120
order does not matter
| Permutations P(10, 3) – order matters | 720 |
| Combinations C(10, 3) – order does not matter | 120 |
| Permutations with repetition (10^3) | 1,000 |
| Combinations with repetition | 220 |
| 10! (n factorial) | 3,628,800 |
Permutations and combinations
Use a permutation when the order of the chosen items matters and a combination when it does not. Both start from the factorial, n! = n × (n − 1) × … × 1.
nPr = n! ÷ (n − r)! · nCr = n! ÷ (r! × (n − r)!)
Example: choose 3 from 10. nPr = 10 × 9 × 8 = 720. nCr = 720 ÷ 3! = 720 ÷ 6 = 120.
Which one do I need?
| Situation | Use |
|---|---|
| Gold, silver, bronze from 10 runners | Permutation (720) |
| A team of 3 from 10 people | Combination (120) |
| 3-digit code from digits 0–9, repeats allowed | nr = 1,000 |
| 3 scoops from 10 flavours, repeats allowed | C(n + r − 1, r) = 220 |
Tips
- 0! is defined as 1, so C(n, 0) = C(n, n) = 1.
- r cannot exceed n when items cannot repeat.
- Results are exact; very large numbers are shown in scientific form.
Frequently asked questions
What is the difference between a permutation and a combination?
A permutation counts arrangements where order matters (a race podium); a combination counts selections where it does not (a committee). Every combination yields r! permutations.
How many ways can 5 people be arranged in a line?
That is 5P5 = 5! = 120 arrangements.
What does nCr = nC(n−r) mean?
Choosing r items to take is the same as choosing n − r to leave behind, so C(10, 3) and C(10, 7) are both 120.
How are lottery odds calculated?
Pick 6 numbers from 49 and the number of tickets is C(49, 6) = 13,983,816, so one ticket has a 1 in 13,983,816 chance.
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