Factorial and binomial topics

{| class="wikitable" style="margin:0 0 0 1em; text-align:right; float:right;" |+ Selected factorials; values in scientific notation are rounded |- ! n ! n! |- | 0 || 1 |- | 1 || 1 |- | 2 || 2 |- | 3 || 6 |- | 4 || 24 |- | 5 || 120 |- | 6 || 720 |- | 7 || |- | 8 || |- | 9 || |- | 10 || |- | 11 || |- | 12 || |- | 13 || |- | 14 || |- | 15 || |- | 16 || |- | 17 || |- | 18 || |- | 19 || |- | 20 || |- | 25 | style="text-align:left" | |- | 50 | style="text-align:left" | |- | 52 | style="text-align:left" | |- | 70 | style="text-align:left" | |- | 100 | style="text-align:left" |

algebraic expansion of powers of a binomial

triangular array of the binomial coefficients in mathematics

thumb|120 px|According to the first meaning of permutation, each of the six rows is a different permutation of three distinct balls|alt=The six different possible ways to order three balls of different colors: (red, green, blue), (red, blue, green), (green, red, blue), (green, blue, red), (blue, red, green), and (blue, green, red). In mathematics, a permutation of a set can mean one of two different things: an arrangement of its members in a sequence or linear order, or the act or process of changing the linear order of an ordered set.

probability distribution

discrete probability distribution

figurate number

polynomial with two terms

family of positive integers that occur as coefficients in the binomial theorem

recursive integer sequence

necessary and sufficient condition for a number to be prime

fractal composed of triangles

probability distribution

discrete probability distribution

In mathematics, and more particularly in number theory, primorial, denoted by "p_{n}\#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers.

probability distribution

distributions defined on [0, 1] in terms of two positive parameters

In 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. As a consequence, an infinite number of multisets exist that contain only elements and , but vary in the multiplicities of their elements: The set contains only elements and , each having multiplicity 1 when is seen as a multiset. In the multiset , the element has multiplicity 2, and has multiplicity 1. In the multiset , and

discrete analog of a derivative

theorem about how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem to polynomials

generalization of the binomial distribution

prime numbers of the form n!±1

mathematical theorem of binomial coefficients

important functions in combinatorics

Taylor series

Type of prime number

the Diophantine problem of finding an integer, whose factorial plus one is a perfect square

product of all the integers from 1 up to the integral input of the function that have the same parity as this input

special function defined by a hypergeometric series

mathematical expression

number theory expression

method for polynomial interpolation over a set of points and corresponding derivatives

number of ways to partition a set of n objects into k non-empty subsets

transformation of a mathematical sequence

theorem

mathematical functions

number of permutations of the numbers from 1 to n in which m elements are greater than the previous element

number of permutations of n elements with k disjoint cycles

type of prime number

mixed radix numeral system adapted to numbering permutations; represents a number as a×0! + b×1! + c×2! + ⋯

three-dimensional arrangement of the trinomial numbers, which are the coefficients of the trinomial expansion and the trinomial distribution

theorem

theorem

sequence of numbers ((2n) choose (n))

numbers giving a solution to several counting problems in combinatorics

conjecture in combinatorial number theory

In mathematics, and more specifically number theory, the hyperfactorial of a positive integer n is the product of the numbers of the form x^x from 1^1 to

family of power series in mathematics

In mathematics, and more specifically number theory, the superfactorial of a positive integer n is the product of the first n factorials. They are a special case of the Jordan–Pólya numbers, which are products of arbitrary collections of factorials.

expectation or average of the falling factorial of a random variable

family of polynomials

The discrete probability distribution of a sum of independent Bernoulli trials that are not necessarily identically distributed

recursive mathematical formula

mathematical sequence

formula in mathematics

theorem on the largest antichain of sets

function that maps an integer 𝑛 to the smallest integer whose factorial is a multiple of 𝑛

mathematical concept

recurrence relations of binomial coefficients in Pascal's triangle

mathematics concept