combinatorics

Examples of combinatorics in the following topics:

• Counting Rules and Techniques

  • Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.
  • Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.
  • Combinatorial techniques are applicable to many areas of mathematics, and a knowledge of combinatorics is necessary to build a solid command of statistics.
  • Aspects of combinatorics include: counting the structures of a given kind and size, deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria.

• Total Number of Subsets

  • These numbers also arise in combinatorics, where $n^b$ gives the number of different combinations of $b$ elements that can be chosen from an $n$-element set.

• Power Series

  • These power series arise primarily in real and complex analysis, but also occur in combinatorics (under the name of generating functions) and in electrical engineering (under the name of the $Z$-transform).

• Theoretical Probability

  • Calculating the number of ways that certain patterns can be formed is part of the field of combinatorics.

• Proof by Mathematical Induction

  • The choice between $n=0$ and $n=1$ in the base case is specific to the context of the proof: If 0 is considered a natural number, as is common in the fields of combinatorics and mathematical logic, then $n=0$.

• Permutations

  • The study of permutations generally belongs to the field of combinatorics.

• Binomial Expansions and Pascal's Triangle

  • The binomial coefficient also arises in combinatorics, where it gives the number of different combinations of $b$ elements that can be chosen from a set of $n$ elements.