Permutations Calculator
Count ordered arrangements with nPr.
Results update as you type.
How it works
A permutation is an ordered arrangement of items, and the permutations calculator tells you how many distinct ordered arrangements are possible when you select r items from a set of n distinct items. Order matters in permutations, which is what separates them from combinations: the arrangement AB is counted as different from BA. This calculator uses the formula nPr equals n factorial divided by the factorial of n minus r, and returns the total count. The intuition is straightforward. When filling the first position you have n choices, for the second position n minus one choices remain, for the third n minus two, and so on, until you have filled all r positions. Multiplying those shrinking counts together gives the number of ordered arrangements, which the factorial formula captures compactly. For example, choosing and arranging 2 items from 5 gives 5 times 4, which is 20 permutations. Permutations appear whenever sequence or ranking matters: the number of ways to award gold, silver, and bronze medals, the possible orderings of a passcode, or the seating arrangements around a table. This calculator requires n and r to be whole numbers that are zero or positive, and r cannot exceed n, since you cannot arrange more items than you have. When those conditions are not met the result is undefined. Students studying probability and combinatorics, along with anyone counting arrangements or orderings, use a permutations calculator to avoid tedious multiplication.
nPr = n! ÷ (n − r)!
A permutation counts how many ways you can arrange r items chosen from n distinct items when order matters. The formula n factorial divided by n minus r factorial works because there are n choices for the first position, n minus one for the second, and so on down to n minus r plus one for the last, and multiplying those together is exactly what the factorial ratio computes. Dividing by n minus r factorial cancels the arrangements of the items you did not select.
Source:Combinatorics
We review all our tools from time to time to ensure accuracy.
Worked example
How to use this calculator
- 1
Enter your values
Enter total items (n) and items chosen (r).
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
Permutations (nPr) updates instantly as you type — no button to press.
Frequently asked questions
What is the difference between permutations and combinations?
Permutations count arrangements where order matters, so AB and BA are different outcomes. Combinations count selections where order does not matter, treating AB and BA as the same group. Because of this, there are always at least as many permutations as combinations for the same n and r. Use permutations for ordered situations like rankings or sequences, and combinations for unordered ones like choosing a committee.
What does the exclamation mark mean?
The exclamation mark denotes a factorial, the product of all positive whole numbers up to and including that number. So 5 factorial, written 5!, equals 5 times 4 times 3 times 2 times 1, which is 120. Factorials count the number of ways to arrange a full set of distinct items in order. By convention zero factorial equals one, which keeps the permutation formula consistent even when n equals r.
Why can't r be larger than n?
The value r is how many items you select and arrange from a set of n items, so it makes no sense to choose more items than actually exist. If r were greater than n, the term n minus r would be negative, and factorials of negative numbers are undefined. Physically there is no way to arrange, say, five items drawn from a set of only three. The calculator returns no result in that case.
What is a real-world example of permutations?
Awarding medals in a race is a classic permutation. If eight runners compete for gold, silver, and bronze, the number of possible podium outcomes is eight times seven times six, because order determines who gets which medal. Other examples include the number of ways to set a combination lock, arrange books on a shelf, or order finishers in any ranked competition where position matters.
What happens when r equals n?
When r equals n, you are arranging the entire set, and the formula reduces to n factorial divided by zero factorial. Since zero factorial equals one, the answer is simply n factorial, the total number of ways to order all the items. For example, arranging all 4 items from a set of 4 gives 4 factorial, which is 24. This is the largest permutation count possible for a given n.