How to use (3 steps)
- Select a mode: n^r for ordered sequences, or nHr for unordered selections (both allow repeats).
- Enter n (types) and r (length / picks). Default values show a common PIN example.
- Switch to Approx when Exact is too large to display. Copy the URL to share exactly what you see.
Results
Digits and scientific notation help you understand scale even when the exact integer is too large to display.
History
Growth chart (n fixed, r increases)
Shows how fast counts grow as r increases with n fixed. Blue: n^r. Green: nHr (multichoose).
How to use this calculator effectively
This guide helps you use Permutations & Combinations with Repetition (n^r, nHr) in a repeatable way: set a baseline, change one variable at a time, and interpret the output with clear assumptions before sharing or exporting results.
How it works
The calculator takes your input values, applies a deterministic formula set, and returns output using display rounding only at the final step. This means the tool is best used as a comparison engine: keep one scenario as a reference, then test alternate assumptions so you can quantify how sensitive the final answer is to each input.
When to use
Use this page when you need a fast planning estimate, a classroom sanity check, or a shareable scenario that another person can reproduce from the same parameters. It is especially useful before deeper modeling, because it exposes direction and magnitude quickly without requiring sign-in or setup friction.
Common mistakes to avoid
- Mixing units (for example, percent vs decimal, or monthly vs yearly assumptions).
- Changing multiple fields at once, which makes it hard to explain why results shifted.
- Comparing outputs from different tools without aligning defaults and conventions.
- Reading rounded display numbers as exact values in downstream calculations.
Interpretation and worked example
Run a baseline case first and keep a copy of that output. Next, change one assumption to represent your realistic alternative, then compare the delta in both absolute and percentage terms. If the direction matches your domain intuition and the size of change is plausible, your setup is likely coherent. If not, review units, sign conventions, and hidden defaults before drawing conclusions.
See also
FAQ
What is the difference between permutations and combinations with repetition?
Permutations with repetition count ordered sequences (AB ≠ BA). Combinations with repetition count unordered selections (AB = BA), allowing repeats.
How do you compute nHr?
nHr can be rewritten as a binomial coefficient: nHr = C(n+r-1, r) = C(n+r-1, n-1).
Can r be larger than n?
Yes. With repetition allowed, you can pick the same type multiple times, so r > n is valid.
When should I use Exact vs Approx?
Use Exact for manageable sizes when you need the full integer. Use Approx when the integer is huge; you still see the digit count and scientific notation.
How does this calculator treat 0^0?
This tool defines n^0 = 1 and treats 0^0 as 1, since the empty sequence is counted as one outcome.
How to use Permutations & Combinations with Repetition (n^r, nHr) effectively
What this calculator does
This page is for estimating outcomes by changing inputs in one controlled workflow. The model keeps your focus on variables, not output shape. Start with stable assumptions, then test sensitivity by changing one key input at a time to observe directional impact.
Input meaning and unit policy
Each input has an expected unit and a typical range. For reliable interpretation, check whether you are using the same unit system, period, and base assumptions across all runs. Unit mismatch is the most common source of unexpected drift in numeric results.
Use-case sequence
A practical sequence is: first run with defaults, then create a baseline log, then run one alternative scenario, and finally compare only the changed output metric. This sequence reduces cognitive load and prevents false pattern recognition in early experiments.
Common mistakes to avoid
Avoid changing too many variables at once, mixing incompatible data sources, and interpreting a one-time output without checking robustness. A single contradictory input can flip conclusions, so keep each experiment minimal and document assumptions as part of your note.
Interpretation guidance
Review both magnitude and direction. Direction tells you whether a strategy moves outcomes in the desired direction, while magnitude helps you judge practicality. If both agree, you can proceed; if not, rebuild the baseline and verify constraints before deciding.