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Probability & Counting

Permutation & Combination Calculator

Choose whether order matters and whether repeats are allowed, enter n and r, and get the exact count. Results use whole-number arithmetic, so even very large answers are not rounded.

Updated October 3, 2026Written and reviewed by Mateuss M.Method: how we calculate

Try an example:

Answer720

10P3 = 720

Step-by-step working 2 steps

1
Order matters, no repeats
```
nPr = n! / (n−r)!  =  n × (n−1) × … (r factors)
```
Use permutations when the order of selection changes the outcome (race places, passwords without repeats).
2
Multiply the first r terms
```
10 × 9 × 8 = 720
```

How to use the permutation & combination calculator

Ask: does order matter? Can items repeat?
Choose the matching counting type.
Enter n and r.
Read the exact count and the steps.

Counting formulas

nPr = n! / (n − r)!
nCr = n! / (r!(n − r)!)
Permutations with repetition = nʳ
Combinations with repetition = C(n + r − 1, r)

nPr = nCr × r!, because each selection can be arranged in r! orders.

Worked examples

Each example below is generated by the same calculator you used above, so the working always matches the answer.

10P3

Answer720

10P3 = 720

Step-by-step working 2 steps

1
Order matters, no repeats
```
nPr = n! / (n−r)!  =  n × (n−1) × … (r factors)
```
Use permutations when the order of selection changes the outcome (race places, passwords without repeats).
2
Multiply the first r terms
```
10 × 9 × 8 = 720
```

10C3

Answer120

10C3 = 120

Step-by-step working 3 steps

1
Order does not matter, no repeats
```
nCr = n! / (r!(n−r)!) = nPr / r!
```
Use combinations when you only care which items are chosen (committees, lottery picks).
2
Compute nPr and r!
```
10P3 = 720
3! = 6
```
3
Divide
```
720 ÷ 6 = 120
```

5-card hands from 52

Answer2,598,960

52C5 = 2,598,960

Step-by-step working 3 steps

1
Order does not matter, no repeats
```
nCr = n! / (r!(n−r)!) = nPr / r!
```
Use combinations when you only care which items are chosen (committees, lottery picks).
2
Compute nPr and r!
```
52P5 = 311,875,200
5! = 120
```
3
Divide
```
311,875,200 ÷ 120 = 2,598,960
```

4-digit PIN

Answer10,000

10^4 = 10,000

Step-by-step working 1 steps

1
Order matters, repeats allowed
```
n^r = 10^4 = 10,000
```
Each of the r positions can be filled in n ways, independently.

Common mistakes to avoid

Using permutations for a committee

Choosing Ann then Raj is the same committee as Raj then Ann, so order does not matter — use combinations.

Forgetting that r cannot exceed n

Without repetition you cannot choose more items than you have.

Expanding factorials by hand for large n

Cancel common factors first, which is what the working shows.

Frequently asked questions

What is 0! and why does nC0 equal 1?

0! is defined as 1. There is exactly one way to choose nothing, so nC0 is 1.

How big can n be?

Up to 500. Results are exact integers, shown with thousands separators.