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Permutations with Repetition Calculator

Count ordered arrangements when repetition is allowed — nʳ ways to fill r positions from n choices, as for PINs, passwords and license plates.

n (choices per position)

r (number of positions)

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How it works

arrangements = nʳ (n choices, r positions, repetition allowed, order matters)

When each position is filled independently from the same n options and repeats are allowed, the choices multiply: n ways for the first slot, n for the second, and so on, giving nʳ. This is the counting rule behind PINs, passwords, license plates and any fixed-length string over a fixed alphabet. It differs from ordinary permutations, where each item can be used only once and the count is n!/(n−r)!.

Worked example

A 4-digit PIN uses 10 digits (0–9) in each of 4 positions with repetition, so there are 10⁴ = 10,000 possible PINs.

Sources: Permutation — Wikipedia

Frequently asked questions

How is this different from a normal permutation?

Normal permutations use each item at most once, giving n!/(n−r)! ordered arrangements. With repetition every position can reuse any option, giving nʳ.

Why does order matter here?

Because the positions are distinct — a PIN of 1234 is different from 4321. Each ordered filling of the slots counts as its own arrangement.

How many 6-character passwords from 62 symbols?

With 62 choices (a–z, A–Z, 0–9) in 6 positions, that is 62⁶ = 56,800,235,584 possible passwords.

Related calculators

Permutation and Combination Calculator

Compute nPr and nCr — how many ways to arrange or choose r items out of n.

Binomial Coefficient Calculator (n choose k)

Compute “n choose k” — the binomial coefficient C(n, k) — the number of ways to pick k items from n where order does not matter.

Multinomial Coefficient Calculator

Count the arrangements of a multiset with the multinomial coefficient n! / (k₁! k₂! …) — e.g. the distinct orderings of the letters in MISSISSIPPI.

Factorial Calculator

Compute n! for any whole number up to 170, with digit count and trailing zeros.

This calculator is for educational purposes. Double-check important results before acting on them.