Permutation Calculator

Calculate ordered arrangements of r items from n total. Find permutations with or without repetition for counting problems.

Last updated: March 2026

Total Items (n)

Items to Choose (r)

Allow Repetition

Items can be selected multiple times

Results

Permutations P(10, 3)

720

10! / (10-3)!

Mode

Without Repetition

Order Matters

Yes

What is a Permutation?

A permutation is an ordered arrangement of r items selected from n total items. The key characteristic: order matters! Selecting items A, B, C is different from selecting B, A, C because they're in different positions.

Permutations answer the question: "In how many different ways can I arrange r items from n?" For example, choosing 1st, 2nd, and 3rd place from 5 runners gives 5×4×3 = 60 different podium arrangements. Each different ordering counts as a separate permutation.

There are two types: without repetition (each item used at most once, typical for real selections) and with repetition (items can be reused, like picking colored balls where you replace after each pick). Without repetition uses the formula P(n,r) = n!/(n-r)!, while with repetition uses P(n,r) = n^r.

How to Use This Calculator

Step-by-Step Guide

Choose n and r: n = total items available, r = items to arrange. Example: 5 runners (n=5), picking top 3 (r=3).

Decide on repetition: Can items be reused? Typically no (can't pick same runner for 1st and 2nd place). Check box if your scenario allows repetition.

Apply formula: Without rep: P(n,r) = n!/(n-r)!. With rep: P(n,r) = n^r. Calculator computes automatically.

Interpret result: Count represents total distinct ordered arrangements. Compare to combinations if order doesn't matter (C < P).

Formulas

Without Repetition: P(n,r) = n! / (n-r)!

With Repetition: P(n,r) = n^r

Where n! = n × (n-1) × (n-2) × ... × 2 × 1

Example Calculation

Race Podium Positions

Scenario:

Arranging top 3 positions (1st, 2nd, 3rd) from 10 runners
n = 10 runners, r = 3 positions
No repetition (same runner can't be in multiple positions)

Step 1: Calculate n!
10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800

Step 2: Calculate (n-r)!
(10-3)! = 7! = 5,040

Step 3: Apply formula
P(10,3) = 10! / 7! = 3,628,800 / 5,040 = 720

Alternative calculation:
P(10,3) = 10 × 9 × 8 = 720 (first 3 factors of 10!)

Interpretation:

There are 720 different ways to award 1st, 2nd, and 3rd place from 10 runners. Each arrangement (like A-B-C vs. B-A-C) counts separately because position matters.

Frequently Asked Questions

What is a permutation?

Permutation: ordered arrangement of r items from n total. Order matters! Example: Arranging 3 people from 10 in a line (first, second, third positions). P(10,3)=720. Compare: combinations (order doesn't matter).

Difference: permutation vs combination?

Permutation (order matters): P(5,2) for 'Top 2 finishers' → AB ≠ BA. Combination (order irrelevant): C(5,2) for 'Choose 2 fruits' → {A,B}={B,A}. Always P ≥ C because P counts all orderings.

Without repetition formula?

P(n,r) = n! / (n-r)! → 'n ways to arrange all, divide by (n-r)! for remaining items.' Shortcut: P(n,r) = n × (n-1) × ... × (n-r+1) [r consecutive factors]. Example: P(5,2) = 5×4 = 20.

With repetition formula?

P(n,r) = n^r → 'Each of r positions can have any of n choices, independently.' Example: 3-digit PIN from 10 digits (0-9) = 10³ = 1,000. Items can repeat: 111, 222, etc. allowed.

Real-world applications?

No rep: Seating arrangements, race finishes (1st/2nd/3rd), lock codes without repeats. With rep: Color sequences allowing duplicates, PIN codes, playlist orderings with song repeats, license plates.

When r > n (no repetition)?

Impossible! Can't arrange 5 items from only 3 total. Formula undefined (factorial of negative number). With repetition, this is allowed: P(3,5) = 3⁵ = 243.

Why does order matter in perms?

P(3,2) has positions [first, second]. AB ≠ BA because different people occupy different seats/positions. Combinations would count {A,B}=1 way regardless. Use perms when rank/position matters (medals, seating).

Factorial growth impact?

10! = 3.6M, 20! ≈ 2.4×10^18 (massive). P(20,5) = 1.86M vs P(20,3) = 6,840. With repetition: 10⁵ = 100K (grows polynomially by power). Without repetition explodes faster for large r.