Statistics

Permutations with Repetition Calculator

Computes n raised to the power r, the number of ordered arrangements of length r when each of the n item types can be reused as often as needed.

What this calculator does

The Permutations with Repetition Calculator estimates your result instantly from the values you enter, using standard statistics formulas. Computes n raised to the power r, the number of ordered arrangements of length r when each of the n item types can be reused as often as needed. Calculations run in your browser, so nothing you type is sent to a server, and the output is an estimate rather than a certified or official figure.

Permutations with Repetition Calculator

Result

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Permutations (nʳ)

About Permutations with Repetition Calculator

This calculator counts permutations with repetition: ordered arrangements of length r drawn from n item types, where each type may be used more than once. Because every position has n independent choices, the count is n^r.

Enter the number of item types and how many positions you are filling to get the total.

Formula & Mathematical Calculation

Permutations (nʳ) = f(Types of items (n), Items chosen (r))

Calculated according to standard stats formulas implemented in the calculator core.

Calculation Example

With 5 item types filling 3 positions: n^r = 5³ = 5 × 5 × 5 = 125 arrangements.

Key Benefits

  • Counts arrangements where items can repeat, unlike standard permutations
  • Applies the simple n^r rule for any number of positions
  • Flags results that grow too large to represent

How to Use this Calculator

  1. Enter types of items (n)
    Type how many distinct item types are available into the Types of items (n) field, for example 5.
  2. Enter items chosen (r)
    Type how many positions you are filling into the Items chosen (r) field, for example 3.
  3. Calculate
    Select Calculate to see the number of arrangements, n raised to the power r.

Assumptions & Limitations

What the calculation assumes

  • Every value you enter is treated as an exact, known amount.
  • Any rate or interval you enter is assumed to hold for the whole period unless the form asks for it separately.
  • Results use standard published formulas and account only for the values entered above.
  • Rounding is applied for display; the underlying calculation uses your full precision.
  • Results are returned in the same units you enter, unless the tool states the output unit.

What it does not cover

  • Permutations with Repetition Calculator produces a calculated result for reference and learning, not a certified or official measurement.
  • Real-world outcomes differ when tolerances, rounding rules or local standards apply.
  • Definitions and standards vary by region and change over time; verify the convention that applies to your case.
  • Figures are only as accurate as the inputs you supply — check them before acting on the result.
  • For work where the result carries real consequences, confirm it against an authoritative source.

Frequently Asked Questions

Here items can repeat, so each of the r positions has all n choices, giving n^r. A standard permutation without repetition uses n! / (n - r)!.

n is the number of item types available and r is the number of positions being filled; the result is n to the power r.

If n^r grows beyond what can be represented, the calculator shows an error that the result is too large.

Free to use, no account needed. Calculations run in your browser, so nothing you type is sent to a server. Results are estimates — how we build and check our calculators.