Permutation and combination calculator
nCr, nPr and factorials with exact answers, including choices with repetition.
- ๐บ๐ธ US federal & state rules
Combinations nCr
120
10C3 = 10! / (3! ร 7!)
- Digits
- 3
- Scientific notation
- 120
- Order matters (10P3)
- 720
This is an example. Enter your numbers above and press the button to see your own result.
Show the math
| Formula | n! / (r! (n - r)!) |
|---|---|
| Same as nPr รท r! | 720 รท 6 |
What to do next
How it works
nPr multiplies the top r numbers: n ร (n - 1) ร ... ร (n - r + 1). nCr divides that by r!, because each group of r items can be ordered r! ways. With repetition, permutations are n^r and combinations are (n + r - 1)C r.
All math uses exact big integers, so results with hundreds of digits are not rounded. Long results show the first and last 15 digits, with the full number available to copy.
Things to keep in mind
Pick the mode by asking two questions: does order matter, and can an item be used more than once? A 4-digit PIN is order matters with repetition (10^4 = 10,000).
Frequently asked questions
What is the difference between a permutation and a combination?
In a permutation the order counts, in a combination it does not. Picking 3 of 10 people gives 720 ordered lineups (10P3) but only 120 groups (10C3).
What is the formula for nCr?
nCr = n! / (r! (n - r)!). For 10C3 that is 3,628,800 / (6 ร 5,040) = 120.
How many Powerball white ball combinations are there?
Choosing 5 of 69 balls gives 69C5 = 11,238,513 combinations, before the separate red Powerball.
Why is 0! equal to 1?
There is exactly one way to arrange nothing. It also keeps formulas like nCn = n! / (n! ร 0!) = 1 working.