> For the complete documentation index, see [llms.txt](https://calctree.gitbook.io/docs/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://calctree.gitbook.io/docs/calculations/creating-calculations/maths-equations/native-functions/probability-and-combinatorics-functions.md).

# Probability & Combinatorics functions

CalcTree includes built-in support for common statistical counting functions and probability-related tools via [MathJS](https://mathjs.org/docs/reference/functions.html#probability). These functions help with uncertainty analysis, risk assessment, and combinatorial calculations.

### Combinatorics Functions

These functions return counts of arrangements, selections, and partitions.

| Function                    | Description                                             | CalcTree Example                  |
| --------------------------- | ------------------------------------------------------- | --------------------------------- |
| `factorial(n)`              | n! — factorial of a number                              | `factorial(5)` → `120`            |
| `combinations(n, k)`        | Ways to choose k unordered items from n (no repetition) | `combinations(5, 2)` → `10`       |
| `combinationsWithRep(n, k)` | Choose k unordered items with repetition                | `combinationsWithRep(3, 2)` → `6` |
| `permutations(n [, k])`     | Ordered subsets of length k (default is n!)             | `permutations(4, 2)` → `12`       |
| `multinomial([a1, a2, …])`  | Ways to divide a set into subgroups of given sizes      | `multinomial([2, 1])` → `3`       |
| `bellNumbers(n)`            | Count of set partitions                                 | `bellNumbers(3)` → `5`            |
| `catalan(n)`                | Number of valid bracketings, rooted trees, etc.         | `catalan(3)` → `5`                |
| `composition(n, k)`         | Ways to write n as a sum of k positive integers         | `composition(5, 2)` → `4`         |
| `stirlingS2(n, k)`          | Stirling numbers of the second kind — set partitions    | `stirlingS2(4, 2)` → `7`          |

### Probability Functions

These functions support basic probability distributions and randomness.

| Function              | Description                                           | CalcTree Example                       |
| --------------------- | ----------------------------------------------------- | -------------------------------------- |
| `gamma(n)`            | Gamma function (generalization of factorial)          | `gamma(6)` → `120`                     |
| `lgamma(n)`           | Logarithm of gamma function                           | `lgamma(6)` → `~4.787...`              |
| `kldivergence(x, y)`  | Kullback-Leibler divergence between two distributions | `kldivergence([0.4, 0.6], [0.5, 0.5])` |
| `random()`            | Random number between 0 and 1 (uniform)               | `random()` → varies                    |
| `random(min, max)`    | Random number in given range                          | `random(10, 20)` → varies              |
| `randomInt(min, max)` | Random integer in range                               | `randomInt(1, 6)` → like a dice roll   |
| `pickRandom(arr)`     | Randomly select one or more values from a list        | `pickRandom([1, 2, 3])` → `2`          |

### Notes for CalcTree

* You can use `random()` in param formulas, but values will refresh on each evaluation.
* Combinatoric functions return exact values, useful for reliability and arrangement modeling.
* Use `gamma(n + 1)` as an alternative to `factorial(n)` for non-integers.

{% hint style="info" %}
📘 **Looking for more functions?**\
CalcTree’s expression engine is powered by [MathJS](https://mathjs.org/docs/reference/functions.html).\
For a full list of available functions, visit the [MathJS Function Reference](https://mathjs.org/docs/reference/functions.html).\
Most functions listed there are supported in CalcTree unless otherwise noted.
{% endhint %}