Permutation and Combination Tool
Explore permutations and combinations using the number of available items, selection size, and repetition rules. Compare four counting methods, inspect logarithmic visualisations, examine nearby selection sizes, and export results in multiple formats directly from your browser.
Same n and r, under each counting rule. Bars show log₁₀(count + 1) so very large counts remain comparable.
| Method | Count | Log₁₀(count) |
|---|
Select a row to set r and recalculate. The highlighted row matches the current result.
| r | Count | Relative scale |
|---|
Representative selections, labelled A, B, C…
Formula and interpretation
Table of Contents
Highlights
- Four counting modes covering permutations, combinations and repetition
- Exact integers for supported sizes and scientific notation for large counts
- Side-by-side comparison chart with logarithmic scaling
- Interactive table for exploring nearby selection sizes
- Export results as CSV, JSON, TXT or SVG
How Permutation and Combination Tool Works?
Counting depends on whether the order of chosen items matters and whether an item can be selected more than once. Enter n, the number of distinct items available, and r, the number selected. Permutations count ordered arrangements; combinations count unordered selections.
| Counting rule | When to use it | Formula |
|---|---|---|
| Permutation, no repetition | Order matters; each item appears at most once | n!/(n−r)! |
| Permutation, repetition | Order matters; items may recur | nr |
| Combination, no repetition | Order does not matter; each item appears at most once | n!/[r!(n−r)!] |
| Combination, repetition | Order does not matter; items may recur | (n+r−1)!/[r!(n−1)!] |
Here, factorial means multiplying all positive whole numbers up to that number, with 0! = 1. For example, selecting three different items from ten without regard to order gives:
The result is shown as a count, alongside its formula and base-10 logarithm. A comparison table and logarithmically scaled bar chart show all four counting rules for the same n and r. Select a row in the nearby-size table to explore a different r. Representative letter-labelled outcomes appear for small inputs. Exact, automatic and scientific display options are available, along with CSV, JSON, TXT and SVG exports.
Practical Use Cases
Students: Compare ordered arrangements and unordered selections to see why the same values of n and r produce different counts.
Event organisers: Count possible prize rankings when winners must be distinct, or form groups when their internal order is unimportant.
Product planners: Count configurable selections when customers can choose several options, including cases where repeated choices are permitted.
Software testers: Estimate the number of sequences or selections in a test scenario and compare the impact of allowing repeated values.
Teachers: Use the comparison chart, nearby selection-size table and small letter-labelled examples to demonstrate combinatorial growth.
Accuracy and Limitations
Inputs must be whole numbers from 0 to 100,000, with r ≤ n when repetition is not allowed; selecting a positive number of items from n = 0 is invalid. Counts are computed exactly using integer arithmetic only when n ≤ 2,000, r ≤ 2,000 and the estimated result is within 7,000 decimal digits. Larger cases use logarithmic approximations, so an exact integer may not be available even when that notation is selected. The logarithm is reported to five decimal places, and chart bars use log10(count + 1), not the raw count. Letter-labelled examples are limited to n ≤ 26 and r ≤ 8 and are illustrative rather than exhaustive.
