Combinatorics: basics

In short: Combinatorics deals with how many ways there are to arrange or select objects. It is the foundation of probability theory and many everyday questions such as password security.

In more detail: Central concepts are permutations (arrangements), combinations (selection without order), and variations (selection with order).

In detail

• Permutations: The number of possible orderings of n distinct objects is n! (n factorial).
• Combinations: When selecting k out of n objects without considering order, the binomial coefficient is used.
• Variations: When the order of selection matters, the formula differs from a pure combination.
• With and without repetition: Whether an element can be chosen multiple times fundamentally changes the calculation.
• Password application: The number of possible passwords of a certain length shows how combinatorics directly affects IT security.
• Lottery: The low chance of winning the lottery can be calculated directly via the binomial coefficient.
• Tree diagrams: For smaller problems, tree diagrams help clearly show all possibilities (wahrscheinlichkeitsrechnung-grundlagen, satz-von-bayes).