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In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations).For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange.
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 ...
For example, repeated numbers appearing across different draws may appear on the surface to be too implausible to be by pure chance. For instance, on September 6, 2009, the six numbers 4, 15, 23, 24, 35, and 42 were drawn from 49 in the Bulgarian national 6/49 lottery, and in the very next drawing on September 10th, the same six numbers were ...
Games of chance are also good examples of combinations, permutations, and arrangements, which are met at every step: combinations of cards in a player's hand, on the table, or expected in any card game; combinations of numbers when rolling several dice once; combinations of numbers in lottery and Bingo; combinations of symbols in slots; permutations and arrangements in a race to be bet on and ...
For example, we might be able to speak of a 1 v 1 + a 2 v 2 + a 3 v 3 + ⋯, going on forever. Such infinite linear combinations do not always make sense; we call them convergent when they do. Allowing more linear combinations in this case can also lead to a different concept of span, linear independence, and basis.
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.
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: (,) = (,) (,) = _! =!
Take for example the group of pairs, adding each component separately modulo some . By omitting one of the components, we suddenly find ourselves in Z n {\displaystyle \mathbb {Z} _{n}} (and this mapping is obviously compatible with the respective additions, i.e. it is a group homomorphism ).