factorial wikipedia - EAS

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  1. From Simple English Wikipedia, the free encyclopedia The factorial of a whole number n, written as n! or n, is found by multiplying n by all the whole numbers less than it. For example, the factorial of 4 is 24, because 4 × 3 × 2 × 1 = 24. Hence one can write 4! = 24.
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    What is a factorial in math?
    Besides nonnegative integers, the factorial can also be defined for non-integer values, but this requires more advanced tools from mathematical analysis . One function that fills in the values of the factorial (but with a shift of 1 in the argument), that is often used, is called the gamma function, denoted Γ(z).
    en.wikipedia.org/wiki/Factorial
    What is the factorial of 0?
    The factorial of 0 is 1, or in symbols, 0! = 1. There are several motivations for this definition: For n = 0, the definition of n! as a product involves the product of no numbers at all, and so is an example of the broader convention that the product of no factors is equal to the multiplicative identity (see empty product ).
    en.wikipedia.org/wiki/Factorial
    What is a double factorial?
    Double factorials are motivated by the fact that they occur frequently in enumerative combinatorics and other settings. For instance, n‼ for odd values of n counts Perfect matchings of the complete graph Kn + 1 for odd n.
    en.wikipedia.org/wiki/Double_factorial
    When were factorials first used?
    Factorials were used to count permutations at least as early as the 12th century, by Indian scholars. Fabian Stedman, in 1677, described factorials as applied to change ringing. After describing a recursive approach, Stedman gives a statement of a factorial (using the language of the original):
    en.wikipedia.org/wiki/Factorial
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    Factorial - Wikipedia

    https://en.wikipedia.org/wiki/Factorial

    In mathematics, the factorial of a non-negative integer , denoted by , is the product of all positive integers less than or equal to . The factorial of also equals the product of with the next smaller factorial: Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the

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    The concept of factorials has arisen independently in many cultures:
    • In Indian mathematics, one of the earliest known descriptions of factorials comes from the Anuyogadvāra-sūtra, one of the canonical works of

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    The factorial function of a positive integer is defined by the product
    If this product formula is changed to keep all but the last term, it would define a product of the same form, for a smaller factorial. This leads to a recurrence relation,

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    The earliest uses of the factorial function involve counting permutations: there are different ways of arranging distinct objects into a sequence. Factorials appear more broadly in many formulas i

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    Several other integer sequences are similar to or related to the factorials:
    Alternating factorial The alternating factorial is the absolute

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    Growth and approximation
    As a function of , the factorial has faster than exponential growth, but grows more slowly than a double exponential function. Its growth rate is similar to , but slower by an exponential factor. One way of approaching this result is

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  4. Giai thừa – Wikipedia tiếng Việt

    https://vi.wikipedia.org/wiki/Giai_thừa

    Trong toán học, giai thừa là một toán tử một ngôi trên tập hợp các số tự nhiên. Cho n là một số tự nhiên dương,"n giai thừa", ký hiệu n! là tích của n số tự nhiên dương đầu tiên.
    n! = 1.2.3....n VD: 4! = 1.2.3.4 = 24 8! = 1.2.3.....7.8 = 40.320
    Đặc biệt, với n = 0, người ta quy ước 0! = 1, đúng theo quy ước của một tích trống. Ký hiệu n! được dùng lần đầu bởi Christian Kramp vào năm 1808. Giai thừa được phổ biến trong nhiều mản…

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  5. Factorial - Simple English Wikipedia, the free encyclopedia

    https://simple.wikipedia.org/wiki/Factorial
    • n! is not defined for negative numbers. However, the related gamma function is defined over the real and complexnumbers (but the integers it is defined over are positive).
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    • Factorial - Wikipedia

      https://ro.wikipedia.org/wiki/Factorial

      35 hàng · Factorial. n! În matematică factorialul unui număr întreg pozitiv n este notat cu n! și …

    • Double factorial - Wikipedia

      https://en.wikipedia.org/wiki/Double_factorial
      • In mathematics, the double factorial or semifactorial of a number n, denoted by n‼, is the product of all the integers from 1 up to n that have the same parity as n. That is, n ! ! = ∏ k = 0 ⌈ n 2 ⌉ − 1 = n ⋯. {\displaystyle n!!=\prod _{k=0}^{\left\lceil {\frac {n}{2}}\right\rceil -1}=n\cdots.} For even n, the double factorial is n ! ! = ∏ k = 1 n 2 = n ⋯ 4 ⋅ 2, {\displaystyle n!!=\prod _{k=1}^{\frac …
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    • Factorial - Wikipedia, la enciclopedia libre

      https://es.wikipedia.org/wiki/Factorial
      • Podemos definir el factorial de un número entero positivo n, expresado n!, como el producto de todos los números enteros positivos menores o iguales que n. 1. n ! = 1 × 2 × 3 × 4 × . . . × ( n − 1 ) × n {\displaystyle n!=1\times 2\times 3\times 4\times ...\times (n-1)\times n} . La multiplicación anterior también se puede representar utilizando el operador productorio: 1. n ! = ∏ k = 1 n k {\di…
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      • Faktoriál – Wikipédia

        https://sk.wikipedia.org/wiki/Faktoriál
        • Faktoriál kladného celého čísla n {\displaystyle n} je definovaný vzťahom: 1. n ! = ∏ k = 1 n k {\displaystyle n!=\prod _{k=1}^{n}k} Pre potreby kombinatoriky je výhodné definovať aj faktoriál nuly. V takom prípade sa definitoricky kladie 0 ! = 1 {\displaystyle 0!=1} .
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        • Факториал — Википедия

          https://ru.wikipedia.org/wiki/Факториал
          • Рекуррентная формула
            Факториал может быть задан следующей рекуррентной формулой: 1. n ! = { 1 n = 0 , n ⋅ ( n − 1 ) ! n > 0. {\displaystyle n!={\begin{cases}1&n=0,\\n\cdot (n-1)!&n>0.\end{cases}}}
          • Комбинаторная интерпретация
            В комбинаторике факториал натурального числа n интерпретируется как количество перестановок (упорядочиваний) множества из n элементов. Например, для множества {A,B,C,D} из 4-х элементов существует 4! = 24 перестановки: Комбинаторная интерпретаци…
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          • Xuất bản: 05/10/2004
        • Factorielle — Wikipédia

          https://fr.wikipedia.org/wiki/Factorielle
          • Soit nun entier naturel. Sa factorielle est formellement définie par : 1. n ! = ∏ 1 ⩽ i ⩽ n i = 1 × 2 × 3 × … × ( n − 1 ) × n . {\displaystyle n!=\prod _{1\leqslant i\leqslant n}i=1\times 2\times 3\times \ldots \times (n-1)\times n.} Le tableau de droite donne les premières factorielles ; par exemple, on a : 1. 1! = 1 2. 2! = 1 × 2 = 2 3. 3! = 1 × 2 × 3 = 6 4. 4! = 1 × 2 × 3 × 4 = 24 5. 10! = 1 × 2 × 3 × 4 × …
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          • แฟกทอเรียล - วิกิพีเดีย

            https://th.wikipedia.org/wiki/แฟกทอเรียล

            22/05/2005 · ในทาง คณิตศาสตร์ แฟกทอเรียล ( อังกฤษ: factorial) ของ จำนวนเต็มไม่เป็นลบ n คือ ผลคูณ ของจำนวนเต็มบวกทั้งหมดที่น้อยกว่าหรือเท่ากับ n ...

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