Permutation Calculator

A permutation calculator computes the number of ways to arrange and order items from a group where order matters. Our nPr calculator uses the permutation formula P(n,r) = n! / (n-r)! to give instant results with step-by-step explanations.

To calculate permutations, use the formula nPr = n! / (n - r)!, where n is the total items and r is the number arranged. For example, P(10,3) = 720. Our online permutation calculator does this automatically with detailed steps.

In permutations, order matters — AB and BA are different. In combinations, order does not matter — {A,B} equals {B,A}. Use our permutation calculator for ordered arrangements, or our combination calculator for unordered selections.

The permutation formula is P(n,r) = n! / (n-r)!, also written as nPr. This counts the number of ways to arrange r items selected from a set of n items where the order of selection matters.

When repetition is allowed, the formula simplifies to n^r, because each of the r choices has n available options. For example, a 4-digit PIN where digits can repeat gives 10^4 = 10,000 permutations.

Permutations are used in cryptography (password strength), probability (lottery odds), scheduling (race finishing orders), genetics (gene sequencing), and anywhere the order of selection matters.

Yes, this free permutation calculator is 100% free with no signup required. Calculate nPr permutations online with step-by-step results instantly.

No. Without repetition, you cannot select more items than you have in the pool. Therefore, r must be less than or equal to n. With repetition allowed, r can exceed n.

nPr stands for "n permutations of r" or "n taken r at a time." It represents the number of ways to arrange r items chosen from a set of n distinct items where order matters.

5P3 = 5! / (5-3)! = 5! / 2! = 60. There are 60 unique ways to arrange 3 items selected from a group of 5. Our nPr calculator computes this instantly.

P(26, 4) = 26! / 22! = 26 × 25 × 24 × 23 = 358,800 permutations. Our permutation calculator with steps shows the full calculation.

0! = 1 by mathematical convention. This ensures the permutation formula works correctly for edge cases, such as when r = n (all items selected).

Permutations count ordered arrangements while combinations count unordered ones. The relationship is P(n,r) = C(n,r) × r!, where C(n,r) is the binomial coefficient.

In circular permutations, arrangements that can be rotated into each other are considered the same. The formula is (n-1)!, which accounts for rotational equivalence.

When items repeat, the formula becomes n! / (n1! × n2! × ... × nk!), where n1, n2, etc. are the counts of each repeated item. This avoids counting duplicate arrangements.

With repetition, each position has n choices regardless of previous selections, giving n^r total permutations. Without repetition, each position has progressively fewer choices.

This calculator handles values up to n = 170, beyond which JavaScript floating-point arithmetic reaches infinity. Results for large values are displayed in scientific notation.

Yes! The permutation formula works regardless of what the items represent. Whether arranging letters, digits, objects, or people, the mathematical calculation is identical.

P(7,3) = 7! / 4! = 7 × 6 × 5 = 210 permutations. This means there are 210 different ordered arrangements of 3 items selected from 7.

A derangement is a permutation where no element appears in its original position. The number of derangements of n items is !n = n! × Σ((-1)^k / k!) for k = 0 to n.

Permutations help calculate the number of favorable outcomes in probability problems where order matters. For example, the probability of drawing 3 specific cards in order is 1/P(52,3).

P(n,r) = C(n,r) × r!. This means permutations equal combinations multiplied by the number of ways to arrange the selected items. Permutations always produce larger values.

The permutation formula says: multiply n × (n-1) × (n-2) × ... × (n-r+1). This gives you the number of ways to fill r positions with n available items where order matters.

The Permutation Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Permutation Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.