Combination - Number of Combinations With Repetition

Number of Combinations With Repetition

See also: Multiset coefficient

A k-combination with repetitions, or k-multicombination, or multiset of size k from a set S is given by a sequence of k not necessarily distinct elements of S, where order is not taken into account: two sequences of which one can be obtained from the other by permuting the terms define the same multiset. In other words, the number of ways to sample k elements from a set of n elements allowing for duplicates (i.e., with replacement) but disregarding different orderings (e.g. {2,1,2} = {1,2,2}). If S has n elements, the number of such k-multicombinations is also given by a binomial coefficient, namely by

(the case where both n and k are zero is special; the correct value 1 (for the empty 0-multicombination) is given by left hand side, but not by the right hand side ). This follows from a clever representation of such combinations with just two symbols (see Stars and bars (combinatorics)).

Read more about this topic:  Combination

Famous quotes containing the words number of, number, combinations and/or repetition:

    “If we remembered everything, we should on most occasions be as ill off as if we remembered nothing. It would take us as long to recall a space of time as it took the original time to elapse, and we should never get ahead with our thinking. All recollected times undergo, accordingly, what M. Ribot calls foreshortening; and this foreshortening is due to the omission of an enormous number of facts which filled them.”
    —William James (1842–1910)

    “But however the forms of family life have changed and the number expanded, the role of the family has remained constant and it continues to be the major institution through which children pass en route to adulthood.”
    —Bernice Weissbourd (20th century)

    “You should try to understand every thing you see and hear; to act and judge for yourselves; to remember you each have a soul of your own to account for; M a mind of your own to improve. When you once get these ideas fixed, and learn to act upon them, no man or set of men, no laws, customs, or combinations of them can seriously oppress you.”
    —Jane Grey Swisshelm (1815–1884)

    “I look on trade and every mechanical craft as education also. But let me discriminate what is precious herein. There is in each of these works an act of invention, an intellectual step, or short series of steps taken; that act or step is the spiritual act; all the rest is mere repetition of the same a thousand times.”
    —Ralph Waldo Emerson (1803–1882)