Permutation With Repetition
Calculate permutations with repetition using the n^r formula. Enter n objects and r positions to find total arrangements — free online calculator with charts.
About This Calculator
The Permutation With Repetition Calculator computes the number of ways to arrange n distinct items into r positions when each item can be used more than once. This fundamental counting principle is essential for combinatorics, password analysis, PIN code estimation, and any scenario involving ordered selections with replacement.
The calculation uses the formula P(n,r) = n^r, where n is the total number of distinct items available and r is the number of selections or positions to fill. For example, if you have 10 digits (0-9) and need a 4-digit PIN, the number of possible PINs is 10^4 = 10,000. The formula works by multiplying n by itself r times because each position has n independent choices.
How It Works
Enter the total number of objects (n) and the sample size (r). The calculator instantly computes n^r to give the total permutations with repetition. It also computes the permutation without repetition value (n!/(n-r)!) and the ratio between the two for comparison. The bar and line charts visualize how the number of permutations grows as r increases from 1 to the entered value.
Regional Notes
India (IN): Permutation with repetition problems appear in Indian JEE Main and Advanced mathematics syllabus under permutations and combinations. Common exam problems involve arranging letters with repetition, creating number plates, and forming passwords.
United States (US): Taught in AP Statistics and high school combinatorics. Used in real-world contexts like license plate combinations, lottery probabilities, and password security analysis.
United Kingdom (UK): Covered in A-Level Mathematics under the probability and combinatorics curriculum. Applications include secure PIN generation analysis and counting principles in decision mathematics.
Frequently Asked Questions
What is permutation with repetition?
Permutation with repetition is an arrangement of items where the order matters and items can be repeated. For example, a 4-digit PIN using digits 0-9 allows repetition (e.g., 1224) and the number of possible PINs is 10^4 = 10,000. The formula is n^r, where n is the number of items available and r is the number of positions to fill.
What is the formula for permutation with repetition?
The formula is P(n,r) = n^r, where n is the total number of distinct items and r is the number of selections or positions. This represents the total number of ways to arrange n items into r places when repetition is allowed.
What is the difference between permutation with and without repetition?
Permutation with repetition (n^r) allows items to be reused in arrangements, while permutation without repetition (n!/(n-r)!) requires each item to be used at most once. For example, with n=10 digits and r=4 places: with repetition gives 10^4 = 10,000 PINs; without repetition gives 10×9×8×7 = 5,040 PINs.
How do you calculate permutations with repetition?
To calculate permutations with repetition, use the formula n^r. Multiply n by itself r times. For example, to find how many 4-letter passwords can be formed from 5 letters (A, B, C, D, E) with repetition allowed: 5^4 = 5×5×5×5 = 625 possible passwords.
What are real-world examples of permutation with repetition?
Common examples include: PIN codes (10 digits into 4 places = 10^4), passwords (62 alphanumeric chars into 8 places = 62^8), combination locks (10 digits into 3 places = 10^3 = 1,000), and license plate configurations where letters and digits can repeat.
What is the difference between permutation and combination with repetition?
In permutations with repetition, order matters (n^r). In combinations with repetition, order does not matter and the formula is C(n+r-1,r) = (n+r-1)!/(r!(n-1)!). For example, selecting 3 scoops from 5 ice cream flavors with repetition: permutations give 5^3 = 125 arrangements, while combinations give C(5+3-1,3) = 35 combinations.
How many permutations with repetition are possible for a standard 4-digit PIN?
For a standard 4-digit PIN using digits 0-9 (n=10) with 4 positions (r=4): n^r = 10^4 = 10,000 possible PINs. This includes PINs like 0000, 1234, 9999, etc.
Can permutation with repetition be applied to password strength analysis?
Yes. Password strength is often measured by the total number of possible permutations. An 8-character password using uppercase (26), lowercase (26), digits (10), and special chars (10) gives n = 72 characters, resulting in 72^8 ≈ 722 trillion permutations. Higher n^r means stronger passwords against brute force attacks.