# Combinatorics

<https://en.wikipedia.org/wiki/Combinatorics>

<https://en.wikipedia.org/wiki/Pascal%27s_triangle> <https://en.wikipedia.org/wiki/Pascal%27s_pyramid> <https://en.wikipedia.org/wiki/Pascal%27s_simplex> <https://en.wikipedia.org/wiki/Binomial_theorem> <https://en.wikipedia.org/wiki/Binomial_distribution>

Combinatorics primarily studies *counting*, both, for its own sake and as a tool in exploring certain properties of finite structures.

The scope of combinatorics traverses many subjects, so it might be better described by the problems it addresses:

* **enumeration** (counting) in connection with discrete structures, known as *arrangements* or *configurations*.
* existence of these structures that satisfy certain criteria.
* possibilities of constructing these structures.
* *optimization problem*: finding the best among possible candidates that satisfy some optimality criteria.

Combinatorics are used in algebra, probability theory, topology, geometry. In the late XX century, powerful theoretical methods were developed that helped promote combinatorics into the independent branch of math. Combinatorics is present in CS through graph theory, and it's often used in complexity theory.

## Fields of combinatorics

* Enumerative combinatorics
* Analytic combinatorics
* Partition theory
* Graph theory
* Design theory
* Finite geometry
* Order theory
* Matroid theory
* Extremal combinatorics
* Probabilistic combinatorics
* Algebraic combinatorics
* Combinatorics on words
* Geometric combinatorics
* Topological combinatorics
* Arithmetic combinatorics
* Infinitary combinatorics

## Related fields

* Combinatorial optimization
* Coding theory
* Discrete and computational geometry
* Combinatorics and dynamical systems
* Combinatorics and physics

<https://rosettacode.org/wiki/Combinations_and_permutations> <https://rosettacode.org/wiki/Combinations> <https://rosettacode.org/wiki/Combinations_with_repetitions> <https://rosettacode.org/wiki/Permutations> <https://rosettacode.org/wiki/Permutations_with_repetitions>

## Combinatorics

<https://rosettacode.org/wiki/Combinations>

* Combinations (order unimportant)
  * without replacement
  * with replacement
* Permutations (order important)
  * without replacement
  * with replacement

## Combinations without replacement

* order unimportant
* without replacement

$$
\displaystyle{
^{n} ! \operatorname{C} \_{k} =
\binom {n} {k} =
\binom {n} {n-k} =
\frac {n!} {(n-k)! \ k!} =
\frac
{ n \cdot (n-1) \cdots (n-k+1) }
{ k \cdot (k-1) \cdots 1      }
}
$$

## Combinations with repetitions

* order unimportant
* with replacement
* with repetitions

$$
\displaystyle{
^{n+k-1} ! \operatorname {C} \_{k} =
\binom {n+k-1} {k} = \ \\
\frac {(n+k-1)!} {(n-1)! \ k!}
}
$$

## Permutations

* order important
* without replacement

$$
\displaystyle{
^{n} ! \operatorname {P} \_{k} =
n \cdot (n-1) \cdot (n-2) \cdots (n-k+1)
}
$$

## Permutations with repetitions

* order important
* with replacement
* with repetitions

$$
\displaystyle {
n^{k}
}
$$


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://mandober.gitbook.io/math-debrief/501-number-theory/630-combinatorics/combinatorics.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.