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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 ...
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.
The number associated in the combinatorial number system of degree k to a k-combination C is the number of k-combinations strictly less than C in the given ordering. This number can be computed from C = {c k, ..., c 2, c 1} with c k > ... > c 2 > c 1 as follows.
These combinations (subsets) are enumerated by the 1 digits of the set of base 2 numbers counting from 0 to 2 n − 1, where each digit position is an item from the set of n. Given 3 cards numbered 1 to 3, there are 8 distinct combinations , including the empty set:
Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets S i indexed by the natural numbers, enumerative combinatorics seeks to describe a counting function which counts the number of objects in S n for each n.
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: (,) = (,) (,) = _! =!
No. 32 appeared most often — 173 times — among the first five balls drawn in winning combinations, followed by the No. 39 in 163 combinations, according to data.
Then 1! = 1, 2! = 2, 3! = 6, and 4! = 24. However, we quickly get to extremely large numbers, even for relatively small n. For example, 100! ≈ 9.332 621 54 × 10 157, a number so large that it cannot be displayed on most calculators, and vastly larger than the estimated number of fundamental particles in the observable universe. [9]