Combinatorial number systemIn mathematics, and in particular in combinatorics, the combinatorial number system of degree k (for some positive integerk), also referred to as combinadics, or the Macaulay representation of an integer, is a correspondence between natural numbers (taken to include 0) N and k-combinations. The combinations are represented as strictly decreasingsequencesck > ... > c2 > c1 ≥ 0 where each ci corresponds to the index of a chosen element in a given k-combination.
Binomial coefficientIn mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written (nk){\displaystyle {\tbinom {n}{k}}} or C(n,k){\displaystyle C(n,k)}.
Pascal's triangleIn mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy.
Natural numberIn mathematics, the natural numbers are the numbers used for counting up, starting from 0 or 1. The first few natural numbers are 0 (if included), 1, 2, 3, and so on. The terms positive integers, non-negative integers, whole numbers, and counting numbers are also used. The set of the natural numbers is commonly denoted by a bold N or a blackboard boldN{\displaystyle \mathbb {N} }.
PermutationMathematical version of an order changeAccording to the first meaning of permutation, each of the six rows is a different permutation of three distinct ballsIn mathematics, a permutation is a bijection of a set onto itself. It can be interpreted as a new order of the members of a sequence or linear order, or as the act or process of changing the linear order of an ordered set.
FactorialProduct of numbers from 1 to nIn mathematics, the factorial of a non-negative integern{\displaystyle n}, denoted by n!{\displaystyle n!}, is the product of all positive integers less than or equal to n{\displaystyle n}.Binomial theoremAlgebraic expansion of powers of a binomial111121133114641151010511615201561172135352171{\displaystyle {\begin{array}{c}1\\1\quad 1\\1\quad 2\quad 1\\1\quad 3\quad 3\quad 1\\1\quad 4\quad 6\quad 4\quad 1\\1\quad 5\quad 10\quad 10\quad 5\quad 1\\1\quad 6\quad 15\quad 20\quad 15\quad 6\quad 1\\1\quad 7\quad 21\quad 35\quad 35\quad 21\quad 7\quad 1\end{array}}}The binomial coefficient(nk){\displaystyle {\tbinom {n}{k}}} appears as the kth entry in the nth row of Pascal's triangle (where the top is the 0th row (00){\displaystyle {\tbinom {0}{0}}}).
MultisetIn mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that element in the multiset.
Set (mathematics)In mathematics, a set is a collection of different things; the things are called elements or members of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, functions, or even other sets. Mathematics typically does not define precisely what constitutes a "set" or "collection", because such a definition would have to be in terms of something else previously defined.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.
If and only ifIn logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") states that the truth values of two statements are equal. It is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false.
List of poker handsA royal flush is the best possible hand in many variants of poker.In poker, players form sets of five playing cards, called hands, according to the rules of the game. Each hand has a rank, which is compared against the ranks of other hands participating in the showdown to decide who wins the pot. In high games, like Texas hold 'em and seven-card stud, the highest-ranking hands win. In low games, like razz, the lowest-ranking hands win.*As an Amazon Associate I earn from qualifying purchases.