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Rather, as explained under combinations, the number of n-multicombinations from a set with x elements can be seen to be the same as the number of n-combinations from a set with x + n − 1 elements. This reduces the problem to another one in the twelvefold way, and gives as result
Wayne Blizard traced multisets back to the very origin of numbers, arguing that "in ancient times, the number n was often represented by a collection of n strokes, tally marks, or units." [ 4 ] These and similar collections of objects can be regarded as multisets, because strokes, tally marks, or units are considered indistinguishable.
A k-combination of a set S is a k-element subset of S: the elements of a combination are not ordered. Ordering the k-combinations of S in all possible ways produces the k-permutations of S. The number of k-combinations of an n-set, C(n,k), is therefore related to the number of k-permutations of n by: (,) = (,) (,) = _! =!
The value of L(n,k,p,t) is of interest to both gamblers and researchers, as this is the smallest number of tickets that are needed to be purchased in order to guarantee a prize. The Hungarian Lottery is a (90,5,5, t )-lotto design and it is known that L(90,5,5,2) = 100.
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The definition of the ratio is the number of common bits, divided by the number of bits set (i.e. nonzero) in either sample. Presented in mathematical terms, if samples X and Y are bitmaps, X i {\displaystyle X_{i}} is the i th bit of X , and ∧ , ∨ {\displaystyle \land ,\lor } are bitwise and , or operators respectively, then the similarity ...
Consider the problem of distributing objects given by a generating function into a set of n slots, where a permutation group G of degree n acts on the slots to create an equivalence relation of filled slot configurations, and asking about the generating function of the configurations by weight of the configurations with respect to this equivalence relation, where the weight of a configuration ...
This is used in two distinct senses: either equal values are considered identical, and are simply counted, or equal values are considered equivalent, and are stored as distinct items. For example, given a list of people (by name) and ages (in years), one could construct a multiset of ages, which simply counts the number of people of a given age.