Search results
Results from the WOW.Com Content Network
The twelve problems of counting equivalence classes of functions do not involve the same difficulties, and there is not one systematic method for solving them. Two of the problems are trivial (the number of equivalence classes is 0 or 1), five problems have an answer in terms of a multiplicative formula of n and x, and the remaining five ...
This would have been the first attempt on record to solve a difficult problem in permutations and combinations. [2] The claim, however, is implausible: this is one of the few mentions of combinatorics in Greece, and the number they found, 1.002 × 10 12, seems too round to be more than a guess. [3] [4]
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.
The second part expands on enumerative combinatorics, or the systematic numeration of objects. It was in this part that two of the most important of the twelvefold ways—the permutations and combinations that would form the basis of the subject—were fleshed out, though they had been introduced earlier for the purposes of probability theory.
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 ...
The enumerations of Theorems one and two can also be found using generating functions involving simple rational expressions. The two cases are very similar; we will look at the case when , that is, Theorem two first. There is only one configuration for a single bin and any given number of objects (because the objects are not distinguished).
This would have been the first attempt on record to solve a difficult problem in permutations and combinations. [4] Al-Khalil (717–786), an Arab mathematician and cryptographer, wrote the Book of Cryptographic Messages. It contains the first use of permutations and combinations, to list all possible Arabic words with and without vowels. [5]