What is the formula for permutations with repetition?

nʳ, where n is the number of available choices and r is the length of the sequence being formed. Each of the r positions can independently be any of the n choices.

How is this different from the Permutation Calculator (nPr)?

The Permutation Calculator computes nPr = n!/(n−r)!, which counts ordered arrangements where each item can be used only once. This calculator computes nʳ, where any item can be reused at any position, allowing repeats.

Why is nʳ always larger than nPr for the same n and r?

Because nʳ allows every position to independently choose from all n options (including repeats), while nPr shrinks the available choices by one at each step since items can't repeat — so nʳ ≥ nPr always, with equality only when r ≤ 1.

What's a real example of permutations with repetition?

A 4-digit PIN code where digits 0–9 can repeat has 10⁴ = 10,000 possible codes, since each of the 4 positions independently has 10 possible digits.

Can r be larger than n?

Yes — unlike ordinary permutations (where r can't exceed n), permutations with repetition allow r to be any size, since items can be reused as many times as needed to fill all r positions.

What happens when r = 0?

n⁰ = 1 for any positive n, representing the single 'empty sequence' — there's exactly one way to select nothing.

How large a result can this calculator handle?

It computes the exact BigInt result for outputs up to 1,000 digits; beyond that, it switches to scientific notation to keep the result readable and the computation fast.

Where are permutations with repetition used in real life?

They're used to count PIN codes, license plate combinations, password strength (character-set size raised to password length), and DNA/RNA sequences built from a fixed alphabet of bases.

Permutations with Repetition Calculator — Ordered Selection with Replacement

Our Permutations with Repetition Calculator computes nʳ — the number of ordered sequences of length r you can build from n available choices when repetition is allowed, like PIN codes, license plates, or DNA sequences. This is distinct from the site's Permutation Calculator, which computes nPr for ordered selections without repetition (each item usable only once). Enter n and r to get the exact count instantly, with BigInt precision for large results.

Quick Answer

Permutations with repetition count ordered sequences where items can repeat: nʳ, where n is the number of choices and r is the sequence length. Enter n and r below for the exact count — this is different from ordinary permutations (nPr), which don't allow repeats.

Number of choices (n)

Sequence length (r)

Enter n and r, then click Calculate.

How to Use the Permutations with Repetition Calculator — nʳ Online

1
Enter the number of available choices (n).
2
Enter the length of the sequence you're selecting (r).
3
Click 'Calculate' to get the exact number of possible ordered sequences with repetition allowed.
4
Review the step-by-step formula showing how nʳ was computed.

Why Use Permutations with Repetition Calculator — nʳ Online?

Permutations with repetition answer a subtly different question than ordinary permutations: instead of 'how many ways can I arrange r items chosen once each from n options,' it asks 'how many ordered sequences of length r can I build if I'm allowed to reuse any option as many times as I like' — the exact scenario behind 4-digit PIN codes (10⁴ = 10,000 combinations) or 3-letter license plate prefixes. The formula nʳ is simple, but it's easy to accidentally apply the no-repetition formula (nPr) instead, which gives a very different, smaller answer. This calculator makes the distinction explicit and computes the exact result.

Frequently Asked Questions

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