What Is nPr?
A permutation is an ordered arrangement. The number of permutations of r items chosen from n distinct items is nPr = n! ÷ (n − r)!. Because order matters, picking A then B counts as a different result from picking B then A.
The calculator also shows the relationship nPr = nCr × r!, the number of arrangements when repetition is allowed (nʳ) and the probability that a random ordered pick matches one target arrangement. If order does not matter in your problem, use the nCr or combination calculator instead.
Below you will find the formulas, a variables table, worked examples and answers to common questions such as nPr versus nCr and how many ways to arrange people in a row.
How to Use the nPr Calculator
- Enter n, the number of distinct items available.
- Enter r, how many of them you arrange in order.
- Read P(n, r), and compare with nCr, repetition and circular counts.
- Open the steps to see the product and the check against nCr × r!.
nPr Formula
Permutations
Link with combinations
Permutations with repetition
Circular permutations
With identical items
nPr Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| n | Total distinct items | |
| r | Items arranged | 0 ≤ r ≤ n |
| nPr | Number of permutations | n! ÷ (n − r)! |
| nʳ | Arrangements with repetition | Each position can reuse an item |
nPr Worked Example
With the example values loaded in the calculator above, the steps are:
- Formula: P(n, r) = n! ÷ (n − r)!.
- Cancel: P(10, 4) = 10 × 9 × 8 × 7.
- Result: 5,040.
- With repetition allowed there are n^r = 10^4 = 10,000 arrangements.
Result: P(10, 4) = 10P4 = 5,040.
How to Read a Permutation Count
P(n, r) counts the ordered ways to arrange r items chosen from n. P(10, 4) = 10 × 9 × 8 × 7 = 5,040. If order does not matter, divide by r! to get combinations: C(10, 4) = 210.
The check P = C × r! ties the two together. If items may repeat, the count is nʳ. Arrangements around a circle are (n − 1)!, because rotations of the same circle count as one arrangement.
Types of permutation problems
| Problem type | Formula | Example |
|---|---|---|
| Arrange r of n distinct items, no repeats | n!/(n − r)! | 3 medals among 10 runners: 720 |
| Arrange all n distinct items | n! | 5 books on a shelf: 120 |
| Items can repeat in each position | nʳ | 4-digit PIN: 10⁴ = 10,000 |
| Some items identical | n!/(n₁! n₂! …) | Letters of MISSISSIPPI: 34,650 |
| Around a circle | (n − 1)! | 6 people at a round table: 120 |
Handling restrictions
Many permutation problems add restrictions. To keep two particular items together, treat them as one block, count the arrangements, then multiply by the ways to order inside the block. To keep two items apart, count all arrangements and subtract those where they are together. For “at least one” conditions, use the complement.
Permutations in probability
If every ordered arrangement is equally likely, the chance of a particular arrangement is 1/P(n, r). Guessing a 4-letter code from 26 letters without repeats has a probability of 1/358,800. Permutation counts also appear in the denominators of many card and lottery probabilities when order matters.
Where nPr Is Used
Ordered selections
Count the ways to award first, second and third place.
Passwords and codes
Count ordered codes with no repeated symbol.
Seating
Count seatings, including circular ones.
Probability
Find the chance that a random ordered pick matches a target.
Common nPr Mistakes to Avoid
- Use permutations when order matters and combinations when it does not.
- Without repetition, each choice leaves one fewer item.
- Identical items reduce the count, so divide by the factorials of the repeat counts.
- Factorials grow very quickly, so even modest n gives very large counts.
nPr Terms and Definitions
- Factorial
- n! is the product of all whole numbers from 1 to n.
- Circular permutation
- An arrangement around a circle where rotations are considered identical.
nPr FAQs
What is nPr?
The number of ordered arrangements of r items chosen from n distinct items, equal to n!/(n − r)!.
How do I calculate permutations?
Multiply n by the next r − 1 smaller numbers. For P(10, 4): 10 × 9 × 8 × 7 = 5,040.
What is the difference between permutations and combinations?
Permutations count ordered arrangements; combinations count unordered selections. P(n, r) = C(n, r) × r!.
What if items can repeat?
Then the count is n^r, because each of the r positions can independently be any of the n items.
What is a circular permutation?
An arrangement around a circle where rotations are treated as the same, giving (n − 1)! for n distinct items.
How do I handle identical items?
Divide n! by the factorial of each group of identical items, such as 11!/(4!4!2!) for MISSISSIPPI.