Permutation and Combination Calculator
Permutations count ordered arrangements and combinations count unordered selections. Choosing a president, treasurer and secretary from 10 people is a permutation because the roles differ. Choosing a 3-person committee is a combination because order does not matter. This tool computes both, plus factorial and the versions that allow repetition.
How to use the permutation and combination calculator
- Enter n, the number of items you choose from, and r, the number you choose.
- Read nPr (order matters) and nCr (order does not matter).
- Use the repetition results when items can be picked more than once.
Formula
- n! = n × (n−1) × … × 1
- nPr = n! ÷ (n−r)!
- nCr = n! ÷ (r! (n−r)!)
- With repetition: permutations = nr; combinations = C(n+r−1, r)
Worked examples
A committee
C(10, 3) = 120 different 3-person committees from 10 people. As officer roles, P(10, 3) = 720 arrangements.
Poker
The number of 5-card hands from 52 cards is C(52, 5) = 2,598,960.
Choosing between nPr and nCr
Ask one question: does order matter? If swapping two chosen items gives a different outcome, use permutations. If not, use combinations.
| Situation | Order matters? | Use |
|---|---|---|
| Race podium (1st, 2nd, 3rd) | Yes | nPr |
| Lottery numbers | No | nCr |
| Locker code 4-2-9 | Yes | nPr or n^r if digits repeat |
| Choosing a team | No | nCr |
Probability connection
Combinations give the denominator for many probability problems. The chance of winning a 6-from-49 lottery is 1 in C(49, 6) = 13,983,816. The chance of being dealt a specific poker hand uses C(52, 5) = 2,598,960 as the total.
The relationship between them
nCr = nPr ÷ r!. Dividing by r! removes the different orderings of the same chosen set. For 10 choose 3, P = 720 and 3! = 6, so C = 120.
Handling big numbers
Factorials explode quickly, so hand calculation quickly becomes impractical. Simplify first: C(52, 5) = 52×51×50×49×48 ÷ 120, cancelling before multiplying. This calculator uses exact big-integer arithmetic, so results are never rounded.
Practice problems
Try these yourself first, then check your answer. The answers come from the calculator above.
- How many ways can 3 medals be awarded among 8 runners?
Show answer
Combinations C(8, 3): 56 · Permutations P(8, 3): 336
- How many 4-person teams can be chosen from 12 people?
Show answer
Combinations C(12, 4): 495 · Permutations P(12, 4): 11,880
Common mistakes to avoid
- Using a permutation when order does not matter, which overcounts by a factor of r!.
- Trying r larger than n without repetition, which is impossible.
- Forgetting that 0! = 1.
Frequently asked questions
What is the difference between a permutation and a combination?
In a permutation order matters (ABC ≠ CBA). In a combination it does not (ABC = CBA).
What is 0 factorial?
1, by convention, so that formulas like C(n, 0) = 1 work.
Why are the numbers so big?
Factorials grow extremely fast: 20! is over 2 quintillion. This tool uses exact big-number maths up to n = 500.
When do I use repetition?
When items can be reused, such as PIN codes (10^4 = 10,000) or picking scoops of ice cream.
Reviewed September 26, 2026 by the SolveCalcPro editorial team. Found a mistake? Tell us; see our editorial policy.