Permutation Calculator
Calculate the number of ordered arrangements (permutations) of r items chosen from a set of n distinct items. The formula is nPr = n! / (n−r)!. Perfect for probability, statistics, combinatorics, and everyday counting problems.
What Is a Permutation?
A permutation is an ordered arrangement of items from a set. The order of the items matters — arranging {A, B, C} is different from {C, B, A}. Permutations are used when the sequence or position of items is important.
The number of permutations of r items chosen from a set of n distinct items (without repetition) is given by:
nPr = n! / (n − r)!
Where:
n = Total number of items
r = Number of items being arranged
! = Factorial (product of all positive integers up to that number)
For example, 5P3 means: "How many ways can you arrange 3 items chosen from a set of 5 distinct items?" The answer is 5! / (5−3)! = 120 / 2 = 60 ways.
How Does the Permutation Calculator Work?
The calculator performs the following steps:
- Step 1: Verify that n ≥ r (you can't arrange more items than you have).
- Step 2: Calculate n! (factorial of n).
- Step 3: Calculate (n − r)! (factorial of the difference).
- Step 4: Divide n! by (n − r)! to get the number of permutations.
- Step 5: Display the result with a detailed breakdown.
You can also calculate permutations with repetition (where items can be repeated). In this case, the formula is n^r.
When to Use Permutations
- Race Results: How many ways can 3 runners finish 1st, 2nd, and 3rd out of 10 runners? 10P3
- Passwords: How many 4-digit passwords can be created from 10 digits? 10P4 (without repetition)
- Seating Arrangements: How many ways can 5 people be seated in 3 chairs? 5P3
- Sports Schedules: How many ways can 8 teams be scheduled for 4 games? 8P4
❓ Permutation Calculator FAQ
What is a permutation?
A permutation is an ordered arrangement of items from a set. The order matters — {A, B} is different from {B, A}. Permutations are used when the sequence or position of items is important.
What is the formula for permutations?
The formula for permutations without repetition is nPr = n! / (n − r)!. For permutations with repetition, the formula is n^r.
What is the difference between a permutation and a combination?
The key difference is that order matters in permutations and order does not matter in combinations. For example, arranging 3 letters from A, B, C, D is a permutation (ABC ≠ ACB). Choosing 3 letters for a committee is a combination (ABC = ACB).
What is the difference between permutations with and without repetition?
Without repetition, each item can be used only once (nPr = n! / (n−r)!). With repetition, items can be reused (n^r). For example, 3-digit passwords from 0-9 without repetition = 10P3 = 720, with repetition = 10^3 = 1000.
What is factorial (n!)?
Factorial is the product of all positive integers up to a number. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1.
How do I calculate 5P3?
5P3 = 5! / (5−3)! = 5! / 2! = 120 / 2 = 60. This means there are 60 ways to arrange 3 items from a set of 5.
What is 10P4?
10P4 = 10! / (10−4)! = 10! / 6! = 10 × 9 × 8 × 7 = 5,040. There are 5,040 ways to arrange 4 items from 10.
What is nPr when r = n?
When r = n, nPn = n! / (n−n)! = n! / 0! = n! (since 0! = 1). This is the number of ways to arrange all n items.
What is nPr when r = 0?
nP0 = n! / (n−0)! = n! / n! = 1. There is exactly 1 way to arrange 0 items (the empty arrangement).
What is the maximum value this calculator can handle?
The calculator can handle n up to 100. However, factorials grow very quickly (100! has 158 digits), so results may be displayed in scientific notation for very large numbers.
What is the difference between permutations and arrangements?
They are essentially the same thing. Both refer to ordered selections. "Permutation" is the formal mathematical term, while "arrangement" is often used in everyday language.
How many ways can 5 people be arranged in a line?
This is 5P5 = 5! = 120 ways. All 5 people are being arranged, so r = n.
How many ways can a president, vice president, and secretary be chosen from 10 people?
This is 10P3 = 10! / 7! = 10 × 9 × 8 = 720 ways. Order matters because the positions are different.
How many 4-digit passwords can be made from digits 0-9 without repetition?
10P4 = 10 × 9 × 8 × 7 = 5,040 passwords. Each digit can be used only once.
How many 4-digit passwords can be made from digits 0-9 with repetition?
10^4 = 10,000 passwords. Digits can be repeated.
What is the difference between permutation and factorial?
Factorial (n!) is the number of ways to arrange all n items. A permutation (nPr) is the number of ways to arrange r items chosen from n items. When r = n, they are the same.
How is this calculator useful for probability?
Permutations are used to count the total number of possible outcomes in probability problems where order matters. This helps calculate the numerator and denominator of probability fractions.
Can I use this calculator for permutations of non-distinct items?
This calculator is designed for distinct items. For permutations with identical items, use the formula n! / (n1! × n2! × ...).
How accurate is this permutation calculator?
This calculator uses JavaScript's BigInt for exact integer calculations up to very large numbers, ensuring accurate results without floating-point errors.