Permutation Calculator
Calculate permutations (nPr), circular permutations, and permutations with repetition. Supports large numbers up to 1000! with exact precision.
Permutation (nPr)
Calculate the number of ways to arrange r items from n distinct items
How to Use the Permutation Calculator
Select the type of permutation calculation, enter your values, and click Calculate. The calculator supports four types of permutations with detailed results including the formula used.
Types of Permutations
Standard Permutation (nPr)
The number of ways to arrange r items from n distinct items where order matters. Formula: P(n,r) = n! / (n-r)!. For example, P(5,3) = 60 represents the 60 ways to arrange 3 items from a set of 5.
All Permutations (n!)
When r = n, you are arranging all items. This is simply n factorial. For example, there are 5! = 120 ways to arrange 5 different books on a shelf.
Circular Permutation
Arrangements in a circle where rotations are considered the same. Formula: (n-1)!. For example, seating 6 people at a round table has (6-1)! = 120 distinct arrangements.
Permutation with Repetition
When items can be repeated, the formula is n^r. For example, a 4-digit PIN using digits 0-9 has 10^4 = 10,000 possible combinations.
Common Applications
- Calculating the number of possible passwords or PINs
- Determining race finish orders
- Seating arrangements at tables
- Scheduling and ordering problems
- Lottery probability calculations
Permutations vs Combinations
In permutations, order matters: ABC is different from CBA. In combinations, order does not matter: ABC and CBA represent the same selection. Permutations are always greater than or equal to combinations for the same n and r values. The exact relationship is P(n,r) = C(n,r) × r!, because each unordered selection of r items can be arranged in r! different orders.
Worked Example: Podium Finishes
Suppose 8 runners compete and you want to know how many different gold-silver-bronze podiums are possible. Order clearly matters, so this is P(8,3) = 8! / (8-3)! = 8 × 7 × 6 = 336. Compare that with the number of ways to pick any 3 medalists regardless of order: C(8,3) = 56. The permutation count is exactly 56 × 3! = 56 × 6 = 336, confirming the relationship above.
Quick Reference Values
- P(10,2) = 90 • P(10,3) = 720 • P(10,4) = 5,040
- P(52,5) = 311,875,200 (ordered 5-card deals from a standard deck)
- Circular arrangements of 8 people: (8-1)! = 5,040
- 6-character case-sensitive alphanumeric strings with repetition: 62⁶ ≈ 5.68 × 10¹⁰
Frequently Asked Questions
What is the difference between nPr and nCr?+
nPr counts ordered arrangements while nCr counts unordered selections. For example, picking 3 runners for gold, silver, and bronze from 8 is P(8,3) = 336, but picking any 3 medalists without ranking them is C(8,3) = 56. The two are linked by P(n,r) = C(n,r) × r!.
How large a number can this permutation calculator handle?+
It accepts n up to 1000 and computes results with exact BigInt precision, so nothing is rounded even when the answer has thousands of digits. For very large results the calculator also shows the value in scientific notation so it stays readable.
When should I use circular permutations?+
Use the circular mode when arrangements around a closed loop are considered identical under rotation, such as seating guests at a round table or arranging beads on a ring. The formula is (n-1)! instead of n!, because fixing one item removes the n equivalent rotations. Seating 6 people around a table therefore gives (6-1)! = 120 arrangements, not 720.
What does permutation with repetition mean?+
It counts sequences where the same item may appear more than once, computed as n^r. A 4-digit PIN built from the digits 0-9 has 10⁴ = 10,000 possibilities because each position independently offers 10 choices. Use this mode for passwords, license plates, and codes.
Is this permutation calculator free to use?+
Yes. It is completely free, requires no account or sign-up, and runs entirely in your browser, so your inputs never leave your device. It works on phones, tablets, and desktops alike.
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