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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 ...
If must be injective, then the selection must involve n distinct elements of X, so it is a subset of X of size n, also called an n-combination. Without the requirement, one and the same element of X may occur multiple times in the selection, and the result is a multiset of size n of elements from X, also called an n-multicombination or n ...
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 ...
Furthermore, the order in which the objects are placed in a boxes does not matter, because there cannot be more than one on each box. So, it is a non ordered injective distribution of 3 indistinguishable objects ( k = 3 {\displaystyle k=3} ) into 7 distinguishable boxes ( n = 7 {\displaystyle n=7} ).
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.
Permutations and 2. Combinations, with clear links between the two, and no combined article. The location for the derivation of the 'combination with repetition' formula can reside in either article as long as there is a link from one to the other explaining its location.
Another way to see this is to rename h 1 to h i, where the derangement is more explicit: for any j from 2 to n, P j cannot receive h j. P i receives h 1. In this case the problem reduces to n − 2 people and n − 2 hats, because P 1 received h i ' s hat and P i received h 1 's hat, effectively putting both out of further consideration.