Number of Combinations With Repetition
See also: Multiset coefficientA 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:
“After mature deliberation of counsel, the good Queen to establish a rule and imitable example unto all posterity, for the moderation and required modesty in a lawful marriage, ordained the number of six times a day as a lawful, necessary and competent limit.”
—Michel de Montaigne (15331592)
“I have known a number of Don Juans who were good studs and who cavorted between the sheets without a psychiatrist to guide them. But most of the busy love-makers I knew were looking for masculinity rather than practicing it. They were fellows of dubious lust.”
—Ben Hecht (18931964)
“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 (18151884)
“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 (18031882)