Permutation Calculator (nPr)Calculate the number of permutations (ordered arrangements) of n items taken r at a time.

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Permutation Calculator (nPr)

Calculate the number of permutations (ordered arrangements) of n items taken r at a time.

How to Use

Enter n

Input the total number of items available.

Enter r

Input the number of items to arrange (must be ≤ n).

View Result

See P(n,r) calculated with the formula and step-by-step breakdown.

What Is Permutation Calculator (nPr)?

A permutation is an ordered arrangement of items selected from a larger set. The number of permutations of r items chosen from n items is denoted P(n,r) or nPr, calculated as n!/(n−r)!. Unlike combinations, permutations consider the order of selection — ABC is different from BAC. For example, if you have 10 runners and want to know how many different podium finishes (gold, silver, bronze) are possible, that's P(10,3) = 10!/(10−3)! = 10 × 9 × 8 = 720. Permutations are used in probability, cryptography, password analysis, tournament brackets, and any scenario where the order of selection matters. This calculator uses BigInt arithmetic to handle large values exactly.

Why Use Our Permutation Calculator (nPr)?

Exact BigInt computation for large values
Shows the formula and calculation breakdown
Handles n up to 1000
Clear distinction from combinations (order matters)
Instant results for any valid n and r

Common Use Cases

Probability Problems

Calculate the number of favorable ordered outcomes for probability computations.

Password Analysis

Determine the total number of possible passwords of a given length from a character set.

Race Outcomes

Calculate how many different finish orderings are possible for a group of competitors.

Seat Arrangements

Find the number of ways to arrange people in specific seats.

Technical Guide

The permutation formula P(n,r) = n!/(n−r)! counts the number of ways to arrange r items from a set of n, where order matters. This can be computed as n × (n−1) × (n−2) × ... × (n−r+1), which is a product of r consecutive integers starting from n. The computation uses BigInt to avoid overflow for large factorials. Key identities: P(n,n) = n! (arrange all items), P(n,1) = n (choose one item), P(n,0) = 1 (one way to arrange nothing). The relationship to combinations is P(n,r) = C(n,r) × r!, because each combination can be arranged in r! ways. The number of permutations grows very rapidly with both n and r.

Tips & Best Practices

1
Remember: permutations care about ORDER (unlike combinations)
2
P(n,r) = n × (n-1) × ... × (n-r+1) — a falling product of r terms
3
P(n,n) = n! — arrange all items in every possible order
4
If r > n, the result is 0 — you can't arrange more items than you have
5
For permutations with repetition, use n^r instead

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Frequently Asked Questions

QWhat is the difference between permutations and combinations?

Permutations count ordered arrangements (ABC ≠ BAC), while combinations count unordered selections (ABC = BAC). Permutations always give a larger or equal result.

QWhen should I use permutations?

Use permutations when the order of selection matters — such as rankings, positions, passwords, or sequences.

QWhat if r equals n?

P(n,n) = n!, which is the total number of ways to arrange all n items. For example, P(4,4) = 24 arrangements of 4 items.

QWhat about permutations with repetition?

If items can repeat, the formula is n^r instead of nPr. For example, a 4-digit PIN using digits 0-9 has 10^4 = 10,000 possibilities.

QHow large can n be?

This calculator supports n up to 1000, computing exact results using BigInt arithmetic regardless of how many digits the answer has.

About Permutation Calculator (nPr)

Permutation Calculator (nPr) is a free online tool from FreeToolkit.ai. All processing happens directly in your browser — your data never leaves your device. No registration required. No ads. Just fast, reliable tools.