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Permutations with repetition calculator

Count the ordered arrangements of length r drawn from n items when repetition is allowed — every position can be any of the n choices independently, giving exactly nʳ. This is the arrangement count for PIN codes, passwords, DNA strings and dice rolls, and it differs from ordinary permutations nPr, which forbid reuse. The result is computed exactly with big integers, however large it grows.

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The Permutations with repetition calculator turns n (number of items), r (length of arrangement) into nʳ arrangements, Number of digits, Scientific notation, instantly and for free. For instance, with n (number of items) = 10 and r (length of arrangement) = 4 it returns nʳ arrangements = 10000, Number of digits = 5 and Scientific notation = 1.0000 × 10^4.

How to use it

  1. Enter your values: n (number of items), r (length of arrangement).
  2. Read the result instantly: nʳ arrangements, Number of digits, Scientific notation.

Frequently asked questions

What does the Permutations with repetition calculator actually compute?

It takes n (number of items) and r (length of arrangement) and derives nʳ arrangements, Number of digits and Scientific notation from them. The calculation is live as you type, so the result updates on every change.

What information do I need to provide?

2 values: n (number of items) and r (length of arrangement). Nothing else is required — no account, no file upload.

Can you show a worked example?

With n (number of items) = 10 and r (length of arrangement) = 4, the calculator returns nʳ arrangements = 10000, Number of digits = 5 and Scientific notation = 1.0000 × 10^4. Those figures come from running this exact tool, so you can reproduce them by entering the same values.

What happens if I enter larger values?

It moves a lot. Using n (number of items) = 20 and r (length of arrangement) = 8 instead, Number of digits goes from 5 to 11 — which is why it is worth testing a few scenarios rather than trusting a single figure.

What does it give for smaller values?

Scaled down to n (number of items) = 5 and r (length of arrangement) = 2, Number of digits comes out at 2. The relationship is worth checking at both ends before you rely on a single result.

When would I actually use this?

Counting possibilities before enumerating them: hands in a card game, passwords of a given shape, seatings around a table, or lottery odds.

What is the most common mistake?

Choosing a permutation when order does not matter. Picking three people from ten gives 720 arrangements but only 120 groups — the two differ by a factor of six here, and far more as the numbers grow.

Where do the figures come from, and how current are they?

Counting formulas are exact by definition. Large factorials are computed with arbitrary precision where needed, so a result is not silently rounded into scientific notation.

Further reading

All guides →
ExplainerCombinations vs Permutations: When Order MattersThe difference between combinations and permutations comes down to one question: does order matter? Learn the nCr and nPr formulas with clear worked examples.ExplainerWhat Is a Factorial? n! Explained SimplyA factorial multiplies every whole number down to 1. Learn what n! means, how fast it grows, and why it powers permutations and combinations.ExplainerOrdinal Numbers Explained: 1st, 2nd, 3rdOrdinal numbers show position, cardinal numbers show quantity. Learn the difference and how each language forms ordinals differently.ExplainerPoker Hand Probabilities, Derived Rather Than MemorisedEvery five-card poker probability is a counting argument over 2,598,960 hands, and each one fits on a single line. Here are all nine, with the check that proves them: the counts must sum to exactly C(52,5).ExplainerHow Exponents Work: Powers and the Laws of ExponentsAn exponent is repeated multiplication. Learn the laws of exponents, what negative and fractional powers mean, and why anything to the zero is one.GuideTournament Brackets: Byes, Seeds, and Why the Numbers Must Be Powers of TwoA knockout bracket halves its field every round, so it only closes on a power of two. The bye count, the round count, the seed order and the total number of matches all follow from that one fact — and each is a one-line derivation.