Permutations with Replacement Calculator
Calculate permutations with replacement (n^r) - arrangements where items can be repeated and order matters. Perfect for PIN codes, passwords, and lock combinations.
Results
Permutations
10,000
10^4
Input Values
Number of choices at each position
Length or number of selections
Show P(n,r) without replacement
Results
With Replacement
Arrangements where items can repeat
10,000
10^4
Security Estimate
Time to try all combinations
10 seconds
at 1,000 attempts/second
With Replacement (n^r)
10,000
Without Replacement (n!/(n-r)!)
5,040
Ratio: 1.98x more with replacement
Step-by-Step Calculation
Visual Representation
Each position has 10 choices = 10^4 = 10,000 total
Practical Examples
Click to calculate:
Password Strength Guide
| Character Set | n | 6 chars | 8 chars | 12 chars |
|---|---|---|---|---|
| Digits only (0-9) | 10 | 1M | 100M | 1T |
| Lowercase (a-z) | 26 | 309M | 209B | 95Q |
| Mixed case (a-z, A-Z) | 52 | 19B | 53T | 390Sx |
| Alphanumeric | 62 | 57B | 218T | 3.2Sx |
| All printable | 95 | 735B | 6.6Q | 540Sx |
M = Million, B = Billion, T = Trillion, Q = Quadrillion, Sx = Sextillion
Permutations Comparison
| Type | Order | Replacement | Formula | Result |
|---|---|---|---|---|
| PR (this) | Yes | Yes | n^r | 10,000 |
| P (standard) | Yes | No | n!/(n-r)! | 5,040 |
| CR | No | Yes | C(n+r-1, r) | - |
| C (standard) | No | No | n!/(r!(n-r)!) | - |
Formula Reference
Permutations with Replacement
PR(n, r) = n^r
Order: Yes | Replacement: Yes
Standard Permutations
P(n, r) = n!/(n-r)!
Order: Yes | Replacement: No
Results
Permutations
10,000
10^4
Common Permutation Examples
Quick-start with common scenarios
Practice Permutation Problems
Test your skills with practice problems
Practice with 4 problems to test your understanding.
?How Do You Calculate Permutations with Replacement?
Permutations with replacement count arrangements where items CAN be repeated and order MATTERS. Formula: PR(n,r) = n^r. Example: 4-digit PIN from digits 0-9 = 10^4 = 10,000 possible codes. Each position has n choices, and we have r positions, giving n x n x ... x n = n^r total arrangements.
What are Permutations with Replacement?
Permutations with replacement count the number of ways to arrange r items from n types where each item can be used multiple times and the order matters. The formula PR(n,r) = n^r comes from having n choices for each of r positions. Common examples include PIN codes (each digit 0-9 can repeat), passwords (letters/numbers can repeat), and combination locks (each dial has same options).
Key Facts
- Permutations with replacement allow using the same item multiple times
- Formula: PR(n,r) = n^r (n to the power of r)
- Order MATTERS: 1234 is different from 4321
- Each position has n independent choices
- Used for: PIN codes, passwords, license plates, phone numbers
- 4-digit PIN (0-9): 10^4 = 10,000 combinations
- 6-character password (a-z): 26^6 = 308,915,776 combinations
- Always larger than permutations without replacement when r > 1
Quick Answer
Permutations with replacement count arrangements where items CAN be repeated and order MATTERS. Formula: PR(n,r) = n^r. Example: 4-digit PIN from digits 0-9 = 10^4 = 10,000 possible codes. Each position has n choices, and we have r positions, giving n x n x ... x n = n^r total arrangements.
Frequently Asked Questions
Permutations with replacement count arrangements where items CAN be repeated and order MATTERS. For example, in a 4-digit PIN, each digit can be 0-9 (repeated), and 1234 is different from 4321. Formula: PR(n,r) = n^r.
With digits 0-9 (10 options) and 4 positions: 10^4 = 10,000 possible PINs. Each position has 10 choices, and digits can repeat. At 1 guess per second, it takes ~2.8 hours to try all combinations.
The formula is PR(n,r) = n^r (n to the power of r). This represents n choices for each of r positions, all independent of each other. The formula works because each position multiplies the total by n options.
WITH replacement: items can repeat, formula is n^r. WITHOUT replacement: items cannot repeat, formula is n!/(n-r)!. Example: 4-digit PIN with repeats = 10^4 = 10,000. Without repeats = 10!/6! = 5,040.
Password strength depends on character set (n) and length (r). Lowercase only (26^r): 6 chars = 309M. Add uppercase (52^r): 6 chars = 19B. Add digits (62^r): 6 chars = 57B. Add symbols (95^r): 6 chars = 735B combinations.
Last updated: 2026-01-15
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How this works
Formulas follow standard definitions from the NIST Digital Library of Mathematical Functions and classical textbook derivations. Calculations run entirely in your browser. Where a closed-form solution exists, it is used; where an iterative or numerical method is required, the implementation is named on the page.
Sources
- [1]NIST Digital Library of Mathematical FunctionsAcademicdlmf.nist.govAccessed Apr 21, 2026
Full-stack software engineer specializing in embedded systems, web architecture, and AI/ML. Founder of Practical Web Tools. Built the gesture-controlled drone IP acquired by KD Interactive (Aura Drone, sold on Amazon).
Results
Permutations
10,000
10^4