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Permutations with Repetition Calculator

Calculate the number of ways to arrange items in a sequence when repetition is allowed.

Result

Total Arrangements
64

Formula: nʳ. Unlike a standard permutation (where each item can only be used once), this counts arrangements where any of the n choices can be repeated at every position, like counting all possible 4-digit PIN codes (10 digits, 4 positions, repetition allowed = 10⁴).

About the Permutations with Repetition Calculator

The Permutations with Repetition Calculator counts the number of ways to fill a fixed number of positions when each position can independently be filled with any of a given set of choices, including repeats. It answers questions like how many distinct 4-digit PIN codes exist from 10 digits, or how many 3-symbol sequences can be built from a 4-symbol set, without listing every arrangement by hand.

How It Works

You enter the number of choices available at each position (n) and the number of positions in the sequence (r). Because every position is filled independently and repetition is allowed, the calculator multiplies n by itself r times to get the total count of distinct arrangements.

Total arrangements = n^r, where n is the number of choices per position and r is the number of positions.

Formula & Methodology

Each position in the sequence has n possible values, and choosing one value for a position never removes it from consideration for the next position. By the multiplication principle, the number of ways to fill the first position is multiplied by the number of ways to fill the second position, and so on across all r positions, giving n multiplied by itself r times.

Examples

4-Symbol Code, 3 Positions

With n = 4 choices and r = 3 positions, the calculator returns 64 total arrangements, since 4 times 4 times 4 equals 64.

4-Digit PIN Codes

With n = 10 (digits 0 through 9) and r = 4 positions, the calculator returns 10,000 total arrangements, matching the number of possible 4-digit PIN codes.

Advantages

  • Instantly computes exponential growth that would take many repeated multiplications to work out by hand for larger position counts.
  • Clearly separates this repetition-allowed formula from the factorial-based permutation formula, reducing the chance of applying the wrong one.
  • Useful as a quick sanity check when estimating the size of a code, password, or combination space.

Common Mistakes

  • Applying the factorial permutation formula, n! divided by (n-r)!, instead of n^r, which only applies when repetition is not allowed.
  • Mixing up which number is n (choices per position) and which is r (number of positions), which changes the result significantly when the two values differ.
  • Treating the result as a probability or security strength rating directly, when it is simply a count of distinct possible arrangements.

Edge Cases to Watch For

  • Both n and r must be non-negative; the calculator rejects negative inputs for either field.
  • If r = 0, the result is always 1 regardless of n, since there is exactly one way to fill zero positions.
  • If n = 0 and r is greater than 0, the result is 0, since there are no choices available to fill even one position.
  • Results grow exponentially with r, so even modest inputs like n = 26 and r = 8 produce numbers in the billions, which the calculator still computes directly.

Common Use Cases

  • Security-minded users estimating how many possible combinations exist for a PIN, password, or access code of a given length.
  • Students learning the multiplication principle and combinatorics fundamentals in an introductory statistics or discrete math course.
  • Game or puzzle designers calculating how many unique codes, license plates, or combination-lock settings a system can support.
Written & fact-checked by the Calculateus TeamLast updated August 5, 2026How we verify our formulas

Frequently asked questions

How is this different from the regular Permutation Calculator?

The regular Permutation Calculator counts arrangements where each item is used at most once (like ranking 5 runners in a race), while this calculator counts arrangements where any choice can repeat at every position (like generating a password from a fixed character set) - the two use completely different formulas (n!/(n-r)! versus nʳ) because they answer different questions.

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Conclusion

The Permutations with Repetition Calculator applies the simple n^r formula to count every possible sequence when items can repeat at each position. It offers a quick way to size up a code space or check combinatorics homework without manually multiplying the same number several times over.

References