About this calculator
Enter n (how many items there are) and r (how many you choose) to get the number of combinations (order doesn't matter) and permutations (order matters).
The answers are exact, using whole-number arithmetic, even when they have dozens of digits.
Worked examples
Real numbers, worked out by the same calculator. Press “Use these numbers” to try one above.
Choose 5 cards from 52
- Combinations (nCr)
- 2,598,960
- Permutations (nPr)
- 311,875,200
- 52! (n factorial)
- 8.06581 × 10⁶⁷ (68 digits)
- 5! (r factorial)
- 120
Choosing 5 from 52: there are 2,598,960 combinations (order doesn't matter) and 311,875,200 permutations (order matters).
Show the working
- Permutations: nPr = n! ÷ (n − r)! = 52! ÷ 47! = 311,875,200.
- Combinations: nCr = nPr ÷ r! = 311,875,200 ÷ 120 = 2,598,960.
Choose 3 from 10
- Combinations (nCr)
- 120
- Permutations (nPr)
- 720
- 10! (n factorial)
- 3,628,800
- 3! (r factorial)
- 6
Choosing 3 from 10: there are 120 combinations (order doesn't matter) and 720 permutations (order matters).
Show the working
- Permutations: nPr = n! ÷ (n − r)! = 10! ÷ 7! = 720.
- Combinations: nCr = nPr ÷ r! = 720 ÷ 6 = 120.
Arrange 6 books on a shelf (6 from 6)
- Combinations (nCr)
- 1
- Permutations (nPr)
- 720
- 6! (n factorial)
- 720
- 6! (r factorial)
- 720
Choosing 6 from 6: there are 1 combinations (order doesn't matter) and 720 permutations (order matters).
Show the working
- Permutations: nPr = n! ÷ (n − r)! = 6! ÷ 0! = 720.
- Combinations: nCr = nPr ÷ r! = 720 ÷ 720 = 1.
The formulas
- Factorial: n! = n × (n − 1) × … × 2 × 1 (and 0! = 1)
- Permutations: nPr = n! ÷ (n − r)!
- Combinations: nCr = n! ÷ (r! × (n − r)!)
Permutation or combination?
Ask: does the order matter? Picking a president, vice-president and treasurer from ten people is a permutation (720 ways), because who gets which job matters. Picking three people for a committee is a combination (120 ways), because the same three people are the same committee however you list them.
Why the numbers grow so fast
Factorials explode: 10! is 3,628,800 and 52! has 68 digits. That is why the number of ways to shuffle a deck of cards is effectively unique every time.
Frequently asked questions
How many 5-card poker hands are there?
52C5 = 2,598,960.
What is 0 factorial?
0! is defined as 1.
What is the difference between nPr and nCr?
nPr counts ordered selections, nCr counts unordered ones, so nCr = nPr ÷ r!.
What is the largest n I can use?
n can be up to 500; beyond that the exact numbers become unmanageably large.
Formulas tested against hand-worked answers. Last reviewed 29 September 2026. These calculators do arithmetic only; they are not financial, tax or legal advice.