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Discrete Math

Permutations with Replacement

Permutations with replacement. Free online Permutations with Replacement. Calculate permutations with replacement online — fast, accurate, mobile-friendly, no s

n^r
125

Derivation

  1. ├── 01Givenn = 5, r = 3
  2. ├── 02Formula(t)^(a)
  3. └── 03Compute n^r125
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§01What is

Understanding the Permutations with Replacement

The Permutations with Replacement computes n^r from 2 inputs: n, r. Permutations with replacement.

Quick calculators for the math that shouldn’t need a notepad — instant, accurate, private to your browser. The Permutations with Replacement sits in that toolkit — it permutations with replacement. Enter your numbers above and the result updates instantly; every step of the math is shown in the Derivation panel so you can see exactly how the answer was reached.

§02The Formula

How it’s calculated

(t)^(a)

Where

n
n
r
r
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: n = 5, r = 3.

  1. 01Start by noting the input — n: 5.
  2. 02Start by noting the input — r: 3.
  3. 03Substitute these values into the formula: (t)^(a)
  4. 04Compute n^r: the calculator returns 125.
  5. 05Cross-check the answer by opening the Derivation panel above — every line of math is shown so you can follow the computation end-to-end.
§04Variants

Common Permutations with Replacement Problems

The formula gets rearranged depending on which variable you need. Here are the patterns you’ll run into in the real world — find the one that matches your problem and follow the worked steps.

01 · PATTERN

n halved

n = 2.5 (from 5)

Keep every other input at its default and halve the n. See how n^r responds.

  1. 01New n: 2.5
  2. 02Baseline n^r: 125
  3. 03New n^r: 15.625
  4. 04n^r decreases by 87.5% → use this sensitivity to plan for real-world variation.
02 · PATTERN

n doubled

n = 10 (from 5)

Keep every other input at its default and double the n. See how n^r responds.

  1. 01New n: 10
  2. 02Baseline n^r: 125
  3. 03New n^r: 1000
  4. 04n^r increases by 700% → use this sensitivity to plan for real-world variation.
03 · PATTERN

r halved

r = 1.5 (from 3)

Keep every other input at its default and halve the r. See how n^r responds.

  1. 01New r: 1.5
  2. 02Baseline n^r: 125
  3. 03New n^r: 11.1803
  4. 04n^r decreases by 91.1% → use this sensitivity to plan for real-world variation.
04 · PATTERN

r doubled

r = 6 (from 3)

Keep every other input at its default and double the r. See how n^r responds.

  1. 01New r: 6
  2. 02Baseline n^r: 125
  3. 03New n^r: 15625
  4. 04n^r increases by 12400% → use this sensitivity to plan for real-world variation.
§05FAQ

Frequently asked questions

Yes. The calculator implements the standard formula as documented and returns exact floating-point results. No approximations are used unless noted in the formula.
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