Permutation Calculator (nPr)
Calculate the permutation nPr, the number of ordered arrangements of r items chosen from n. Also shows nCr and n factorial.
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Input
Enter the total count n and the number to choose r to calculate the permutation nPr, the number of ordered arrangements. The combination nCr and n factorial are shown as well.
Integer of 0 or more
Integer from 0 to n
Result
Permutation nPr choosing r equal to 3 from 5
60
Combination nCr with n equal to 5 and r equal to 3
10
Factorial n with n equal to 5
120
Digits of nPr
2
The permutation nPr equals n factorial divided by (n minus r) factorial, and nCr equals nPr divided by r factorial.
How it works
- A permutation nPr is the number of ways to choose r items from n distinct items and arrange them in order, given by nPr equal to n factorial divided by (n minus r) factorial.
- Equivalently, nPr is the product of r consecutive integers starting at n, that is n times (n minus 1) and so on down to (n minus r plus 1).
- A combination nCr counts selections where order does not matter, given by nCr equal to nPr divided by r factorial.
- When r is 0 both nPr and nCr equal 1, and when r equals n the permutation nPr equals n factorial.
- The inputs must satisfy 0 less than or equal to r less than or equal to n, so r cannot exceed n.
- This tool uses big integer arithmetic, so results stay exact even when the number of digits becomes very large.
Frequently asked questions
- What is the formula for the permutation nPr?
- It is nPr=n!÷(n−r)!, equivalently the product of r consecutive integers counting down from n: n×(n−1)×…×(n−r+1). For example, 5P2=5×4=20.
- How is a permutation nPr different from a combination nCr?
- A permutation counts ordered arrangements, so order matters, while a combination nCr ignores order. They are linked by nCr=nPr÷r!, and this tool shows nCr alongside nPr.
- What happens when r is 0 or larger than n?
- When r=0 the result is 1, and when r=n it equals n!. The inputs must satisfy 0≤r≤n, so r larger than n cannot be computed.
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