Updated · Built and checked by the PlinzoTools team, formulas tested against published sources
Calculate nPr -- the number of ordered arrangements of r items from n.
20
A permutation counts the number of ways to arrange r items from a set of n, where order matters (e.g. 1st, 2nd, 3rd place in a race are different arrangements).
Enter n (total items) and r (items chosen, r ≤ n). The number of permutations appears instantly.
nPr = n! / (n-r)!
Use permutations when the order in which you pick items changes the outcome. Awarding gold, silver, and bronze to three of eight competitors is a permutation problem, because giving gold to Alex and silver to Blake is different from giving gold to Blake and silver to Alex. Passwords, PIN codes without repeated digits, seating charts, and ranked lists all count arrangements in which position matters.
A hypothetical club has 8 members and needs to fill three distinct offices: president, secretary, and treasurer. Each person can hold only one office. Enter n = 8 and r = 3.
8P3 = 8 × 7 × 6 = 336
There are 336 ways to fill the offices. You can reason it out directly: 8 choices for president, then 7 remaining for secretary, then 6 for treasurer. The tool multiplies exactly these descending terms, so you get the same answer without computing the large factorials separately.
Both n and r must be whole numbers, neither can be negative, and r cannot exceed n. If you enter a decimal, a negative number, or an r larger than n, the tool shows no result, since you cannot choose more distinct items than exist. Setting r = 0 gives 1, meaning there is exactly one way to arrange nothing, and r = n gives n!, the count of all full orderings.
This calculator assumes each item can be used at most once. If repeats are allowed, the count is different. A 4-digit code using any of 10 digits with repeats permitted has 10 × 10 × 10 × 10 = 10,000 possibilities, which is far more than 10P4 = 5,040. Before you rely on the answer, ask yourself whether the situation really forbids reuse.
The tool multiplies exact integers, so every digit is correct however large the count gets. n can be up to 10,000. Results up to 40 digits are written in full with thousands separators; longer ones are shown in scientific notation with the exact leading digits. For most classroom and everyday counting problems you will see the full number.
Every combination of r items can be arranged in r! different orders. That is why nPr equals nCr multiplied by r!. In the club example above, 8C3 is 56, and 56 × 3! = 56 × 6 = 336, matching the permutation count.
Permutations count ordered arrangements (AB and BA are different); combinations count unordered selections (AB and BA are the same). See the Combination Calculator.
nPn = n! -- every possible ordering of all n items.
Ask whether swapping two selected items creates a different result. If it does, such as first place versus second place, use permutations. If it does not, such as choosing team members, use combinations.
You cannot pick more distinct items than exist in the group without repeating some. Since this tool does not allow repeats, r must be less than or equal to n.
It equals n, since there are n choices for a single position.
That is 10P4 = 10 × 9 × 8 × 7 = 5,040 ways, assuming no one can appear twice.
Reference: NIST Weights and Measures