Permutation with Repetition Calculator

Enter the total number of items (n) and the permutation size (r) to count how many ordered arrangements are possible when repetition is allowed. The Permutation with Repetition Calculator applies the formula P = n^r and returns the total number of permutations along with a breakdown of your inputs. Perfect for password combinations, PIN codes, or any ordered sequence where items can repeat.

How many distinct items are in your set (e.g. 10 digits: 0–9)

How many positions or slots in each arrangement (e.g. a 4-digit PIN)

Results

Total Permutations (n^r)

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Items in Set (n)

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Arrangement Size (r)

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Formula Used

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Permutations for Different Arrangement Sizes

Results Table

Frequently Asked Questions

What is a permutation with repetition?

A permutation with repetition is an ordered arrangement of items from a set where each item can be used more than once. For example, the PIN code 3355 is a permutation with repetition because the digits 3 and 5 each appear twice. This differs from permutations without repetition, where each item can only be used once.

What is the formula for permutation with repetition?

The formula is P = n^r, where n is the total number of distinct items in the set and r is the number of positions in each arrangement. For example, if you have 10 digits (0–9) and want 4-digit PINs, the calculation is 10^4 = 10,000 possible PINs.

How do you calculate permutation with repetition?

Simply raise the number of available items (n) to the power of the arrangement size (r). If you have 26 letters and want 3-character codes, compute 26^3 = 17,576 possible codes. Each position independently allows all n choices, so the total multiplies out as n × n × n... (r times).

How many different 10-character passwords are possible using only alphanumeric characters?

Alphanumeric characters include 26 lowercase letters, 26 uppercase letters, and 10 digits — a total of 62 characters. Using the formula n^r = 62^10, there are 839,299,365,868,340,224 possible 10-character alphanumeric passwords. That's over 839 quadrillion combinations.

What is the difference between permutation with and without repetition?

In permutations without repetition, each item from the set can only appear once per arrangement, and the formula is n! / (n−r)!. In permutations with repetition, items can be reused in any position, and the formula simplifies to n^r. The with-repetition count is always greater than or equal to the without-repetition count.

What is the Cartesian product, and how does it relate to permutations with repetition?

The Cartesian product of r identical sets of n elements produces all ordered r-tuples from those elements — which is exactly the set of all permutations with repetition. The total count of elements in this Cartesian product is n^r, confirming the permutation with repetition formula.

Can n or r be zero in permutation with repetition?

If r = 0, there is exactly 1 permutation — the empty arrangement — since n^0 = 1 for any non-zero n. If n = 0, no items exist so no arrangements are possible (0^r = 0 for r > 0). This calculator requires n ≥ 1 and r ≥ 1 to return meaningful results.

When would I use a permutation with repetition in real life?

Common real-world applications include counting possible PIN codes or passwords (where digits/characters can repeat), determining the number of possible outcomes when rolling a die multiple times, calculating how many license plate combinations exist, and analyzing sequences in genetics or coding theory where symbols can repeat.

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