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It can be used to solve a variety of counting problems, such as how many ways there are to put n indistinguishable balls into k distinguishable bins. [4] The solution to this particular problem is given by the binomial coefficient ( n + k − 1 k − 1 ) {\displaystyle {\tbinom {n+k-1}{k-1}}} , which is the number of subsets of size k − 1 ...
It covers most notably his theory of permutations and combinations; the standard foundations of combinatorics today and subsets of the foundational problems today known as the twelvefold way. It also discusses the motivation and applications of a sequence of numbers more closely related to number theory than probability; these Bernoulli numbers ...
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures.It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science.
In combinatorics, the twelvefold way is a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of counting permutations, combinations, multisets, and partitions either of a set or of a number.
The Dunst Opening is a chess opening in which White opens with the move: . 1. Nc3. This fairly uncommon opening may have more names than any other: it is also called the Heinrichsen Opening, Baltic Opening, Van Geet Opening, Sleipnir Opening, Kotrč's Opening, Meštrović Opening, Romanian Opening, Queen's Knight Attack, Queen's Knight Opening, Millard's Opening, Knight on the Left, and (in ...
The Vienna Game is an opening in chess that begins with the moves: . 1. e4 e5 2. Nc3. White's second move is less common than 2.Nf3, and is also more recent. The original idea behind the Vienna Game was to play a delayed King's Gambit with f4 (the Vienna Gambit), but in modern play White often plays more quietly (for example, by fianchettoing their king's bishop with g3 and Bg2).
Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets S i indexed by the natural numbers , enumerative combinatorics seeks to describe a counting function which counts the number of objects in S n for each n .
Combinations and permutations in the mathematical sense are described in several articles. Described together, in-depth: Twelvefold way; Explained separately in a more accessible way: Combination; Permutation; For meanings outside of mathematics, please see both words’ disambiguation pages: Combination (disambiguation) Permutation ...