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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.
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 ...
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 ...
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 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 ...
Heap's algorithm generates all possible permutations of n objects. It was first proposed by B. R. Heap in 1963. [1] The algorithm minimizes movement: it generates each permutation from the previous one by interchanging a single pair of elements; the other n−2 elements are not disturbed.
An archetypal double counting proof is for the well known formula for the number () of k-combinations (i.e., subsets of size k) of an n-element set: = (+) ().Here a direct bijective proof is not possible: because the right-hand side of the identity is a fraction, there is no set obviously counted by it (it even takes some thought to see that the denominator always evenly divides the numerator).
The conversion can be done via the intermediate form of a sequence of numbers d n, d n−1, ..., d 2, d 1, where d i is a non-negative integer less than i (one may omit d 1, as it is always 0, but its presence makes the subsequent conversion to a permutation easier to describe).