# Combinatorics

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## Area of ​​Expertise - mathematics

Combinatorics is the branch of mathematics that deals with the laws governing the various possible arrangements for a set of given objects (elements).

Typical examples are:

• How many possibilities are there, six numbers from the set of numbers ${1;2;3;⋯;49}$ to select?
• How many ways are there to combine four different amino acids into an oligopeptide that consists of 16 amino acids?
• How large is the number of unique images one $n$-elemental amount on yourself?

As can be seen from the examples, the focus of the combinatorics is the determination of the number of the respective arrangements. It is therefore an important aid in calculating probability.

## Learning units in which the term is discussed

### Combinatorial drug design60 min.

#### PharmacyPharmaceutical chemistryDrug design

The aim of combinatorial design is to generate the largest possible number of potential active ingredients from the combination or permutation of simple basic substances. In contrast to the conventional synthesis of active ingredients, which proceeds in a targeted manner from a few starting materials to one or a few desired end products, combinatorial synthesis tries to achieve the greatest possible variety of synthesis products. The combinatorial drug design thrives on the principle that a large number of testable substances also increases the number of possible new lead structures. A second advantage of the combinatorial method comes in after new lead structures have been found through trial and error. Since it is relatively easy to systematically modify structures in combinatorial synthesis, the chance and the time required to optimize the lead structures that have been found increases increases.

### Combinatorics - an excursus40 min.

#### MathematicsCombinatoricsCombinatorics

Combinatorics is a branch of mathematics that deals with the investigation of the various arrangements of finitely many elements. Other disciplines such as statistics and probability calculus use the methods of combinatorics.

### Evolutive methods45 min.

#### BiochemistryWorking methodsGenetic engineering

Examples of the application of evolutionary methods: phage and ribosome display, Merrifield syntheses.

### Permutations20 min.

#### MathematicsCombinatoricsCombinatorics

A permutation is an arrangement of all elements of a finite set. The learning unit shows how many permutations are possible for a given number of elements. The explanations are part of the combinatorics.

## Video: Combination formula. Probability and combinatorics. Probability and Statistics. Khan Academy (July 2022).

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