Search results
Results from the WOW.Com Content Network
The set of all k-combinations of a set S is often denoted by (). A combination is a selection of n things taken k at a time without repetition. To refer to combinations in which repetition is allowed, the terms k-combination with repetition, k-multiset, [2] or k-selection, [3] are often used. [4]
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 problem of finding a closed formula is known as algebraic enumeration, and frequently involves deriving a recurrence relation or generating function and using this to arrive at the desired closed form. Often, a complicated closed formula yields little insight into the behavior of the counting function as the number of counted objects grows.
The bins are distinguished (say they are numbered 1 to k) but the n objects are not (so configurations are only distinguished by the number of objects present in each bin). A configuration is thus represented by a k-tuple of positive integers. The n objects are now represented as a row of n stars; adjacent bins are separated by bars. The ...
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 algorithm minimizes movement: it generates each permutation from the previous one by interchanging a single pair of elements; the other n−2 elements are not disturbed. In a 1977 review of permutation-generating algorithms, Robert Sedgewick concluded that it was at that time the most effective algorithm for generating permutations by computer.
The formula for permutation with repetitions in its traditional sense is N!/(m1!m2!...mLast!), where N = m1 + m2 + ... + mLast is cardinality of the set (total number of objects), and m1, m2, ... are numbers of identical objects of type 1, 2, ... The formula for combination with repetitions also understands repetitions in the same wrong way ...