What is the formula for permutations with repetition?

The formula is n^r, where n is the number of distinct items to choose from and r is the number of positions to fill. Each position can hold any of the n items independently.

How is this different from permutations without repetition?

Without repetition, each item can be used at most once, giving n!/(n-r)! arrangements. With repetition allowed, each slot is independent, so the count is n^r — always greater than or equal to the without-repetition count.

n^0 = 1 for any positive n (there is exactly one empty arrangement). 0^r = 0 for any positive r (no items to choose from means no arrangements). 0^0 is treated as 1 by convention.

Can n be smaller than r?

Yes — unlike permutations without repetition, n can be less than r because items can repeat. For example, 2 items in 5 positions gives 2^5 = 32 sequences.

What is the largest value this calculator handles?

The tool uses arbitrary-precision integer arithmetic in JavaScript (BigInt) so there is no practical upper limit on the result size, though very large values display in scientific notation for readability.

Permutations With Repetition Calculator

Name: Permutations With Repetition Calculator
Availability: InStock
Author: Nham Vu

Enter n (pool size) and r (positions) to count how many ordered sequences exist when items can repeat.

Inputs

Formula: n^r

n — pool size (number of distinct items)

r — positions (slots to fill)

Quick examples

Result

Enter n and r to see the result

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