Permutation and Combination Calculator
How many ways to arrange or choose r items from n, with or without repetition, exactly for large numbers.
Result
Number of ways: 120
Exact counting while the result fits; flagged where it does not.
- Formula
- C(n, r) = n! ÷ (r! × (n − r)!)
- Precision
- Exact
- n!
- 3,628,800
https://calcgallery.com/statistics/permutation-combination-calculator/
Results follow the mathematical operation and values shown. Check that you selected the intended operation, convention and dataset type before relying on the result. Read the full disclaimer
How to use the Permutation and Combination Calculator
- Say whether order matters, and whether an item can be used more than once.
- Enter how many items there are to choose from, n, and how many are chosen, r.
- Read the number of ways, with the formula used and whether the result is exact.
The four formulas
| Order matters | Order does not matter | |
|---|---|---|
| No repetition | P(n, r) = n! ÷ (n − r)! | C(n, r) = n! ÷ (r! × (n − r)!) |
| Repetition allowed | nʳ | C(n + r − 1, r) |
A permutation counts arrangements, and a combination counts groups. Every combination of r items can be arranged in r! orders, so C(n, r) is P(n, r) divided by r!.
With repetition, a 3-digit code from the digits 0 to 9 has 10 × 10 × 10 = 1,000 possibilities. Three scoops of ice cream from 10 flavours, where two scoops can be the same flavour and the order in the cone does not matter, has C(12, 3) = 220.
Why the page never divides factorials
The formulas are written with factorials, but calculating them that way fails quickly. 171! is too large for a computer’s ordinary numbers, so C(200, 2) would come out as infinity divided by infinity, which is not a number. The true answer is 19,900.
This page multiplies step by step instead, (n − r + 1) ÷ 1 × (n − r + 2) ÷ 2 and so on, and each step is itself a whole number. The result stays exact while it is below 2⁵³, and when it goes above, the page says so rather than showing digits it cannot vouch for.
With your numbers
- C(n, r) = n! ÷ (r! × (n − r)!), with n = 10 and r = 3
- Number of ways = 120
Quick reference
Exact counting while the result fits; flagged where it does not.
| Example | Number of ways |
|---|---|
| Choose 3 of 10 | 120 |
| Arrange 3 of 10 | 720 |
| 3-digit code, digits 0 to 9 | 1,000 |
| 3 scoops from 10 flavours | 220 |
| Poker hand, 5 of 52 | 2,598,960 |
| Lottery, 6 of 49 | 13,983,816 |
| Pairs from 200 | 19,900 |
Common questions about the Permutation and Combination Calculator
- How do I know whether order matters?
- Ask whether swapping two of the chosen items gives a different outcome. For a race podium it does, since first and second are different results, so it is a permutation. For a hand of cards or a committee it does not, so it is a combination. A combination lock is badly named: the order of the numbers matters, so it is really a permutation.
- Why is there exactly one way to choose nothing?
- Because choosing no items is one outcome, the empty choice, not zero outcomes. C(n, 0) = 1 for every n, and it keeps formulas such as the binomial theorem working without special cases.
- Why does the page say the result is approximate?
- Above 2⁵³, about 9 × 10¹⁵, a computer’s ordinary numbers cannot hold every whole number exactly, so the last digits are rounded. The page counts without dividing huge factorials, which keeps everything below that limit exact, and says so when the answer passes it. For very large results it also gives the number in scientific notation.
- What are the odds of winning a 6 from 49 lottery?
- One in C(49, 6) = 13,983,816 for a single line, since the order the balls are drawn in does not matter. That is one of the examples in the table on this page.