Definition
In mathematics, the factorial of a natural number n is the product of the positive integers less than or equal to n. This is written as n! and pronounced ‘n factorial’.
The factorial function is formally defined by
- =1 × 2 × 3 × . . . × (n – 1) × n
- For example,
- 5! = 1 * 2 * 3 *4 * 5 = 120
This definition implies in particular that
- 0! = 1
because the product of no numbers at all is 1. Proper attention to the value of the empty product is important in this case, because
- it makes the above recursive relation work for n = 1;
- many identities in combinatorics would not work for zero sizes without this definition
The factorial of an integer n is denoted by n!. This n! notation was first used by a French mathematician. Who was he?
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Given that n! = n(n-1)(n-2)…(2)(1). Instead of multiplication, there exists an analog series that is defined by addition, that is, f(n) = n + (n-1) + (n-2) + … + 2 + 1. This type of number is known as a/an?
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A prime number that is 1 more or 1 less than the value of a factorial is called a factorial prime. Which of the following numbers is not a factorial prime?
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Factorial operations are used widely in combinations and permutations problems. To find the number of different ways one can select r items from a total of n items, we use the combination formula, which is n C r = (n!)/{[(n-r)!](r!)}. On the other hand, the formula for permutation, given by n P r, is?
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The factorial of 0, namely 0! is 0.
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Factorial operations are used widely in combinations and permutations problems. To find the number of different ways one can select r items from a total of n items, we use the combination formula, which is n C r = (n!)/{[(n-r)!](r!)}. On the other hand, the formula for permutation, given by n P r, is?
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A positive integer that is in the form of n^{(n-1)^[(n-2)^…^{2^(1)}]} is called a/an ?
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Factorials also come in useful in calculus. Some exponential and trigonometry functions can be expressed as a power series of x, as what is stated and defined in which of the following theorems?
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Calculating factorials
The numeric value of n! can be calculated by repeated multiplication if n is not too large. That is basically what pocket calculators do. The largest factorial that most calculators can handle is 69!, because 70! > 10100.
When n is large, n! can be estimated quite accurately using Stirling’s approximation:
The gamma function
The related gamma function Γ(z) is defined for all complex numbers z except for the nonpositive integers (z = 0, −1, −2, −3, …). It is related to factorials in that it satisfies a recursive relationship similar to that of the factiorial function:
- n! = n(n – 1)!
- Γ(n + 1) = nΓ(n)
Together with the definition Γ(1) = 1 this yields the equation
Because of this relationship, the gamma function is often thought of as a generalization of the factorial function to the domain of complex numbers. This is justified for the following reasons.
- Shared meaning—The canonical definition of the factorial function is the mentioned recursive relationship, shared by both.
- Uniqueness—The gamma function is the only function which satisfies the mentioned recursive relationship for the domain of complex numbers and is holomorphic and whose restriction to the positive real axis is log-convex. That is, it is the only function that could possibly be a generalization of the factorial function.
- Context—The gamma function is generally used in a context similar to that of the factorials (but, of course, where a more general domain is of interest).
Multifactorials
A common related notation is to use multiple exclamation points to denote a multifactorial, the product of integers in steps of two (n!!), three (n!!!), or more.
n!! denotes the double factorial of n and is defined recursively by
For example, 8!! = 2 · 4 · 6 · 8 = 384 and 9!! = 1 · 3 · 5 · 7 · 9 = 945. The sequence of double factorials for n = 0, 1, 2,… starts
- 1, 1, 2, 3, 8, 15, 48, 105, 384, 945, 3840, …
Some identities involving double factorials are:
- n! = n!!(n – 1)!!
- (2n)!! = 2nn!
One should be careful not to interpret n!! as the factorial of n!, which would be written (n!)! and is a much larger number.
The double factorial is the most commonly used variant, but one can similarly define the triple factorial (n!!!) and so on. In general, the k-th factorial, denoted by n!(k), is defined recursively as
- <img alt=" n!^{(k)}= \left\{ \begin{matrix} 1,\qquad\qquad\ &&\mbox{if }0\le n
Hyperfactorials
Occasionally the hyperfactorial of n is considered. It is written as H(n) and defined by
For n = 1, 2, 3, 4,… the values of H(n) are 1, 4, 108, 27648,…
The hyperfactorial function is similar to the factorial, but produces larger numbers. The rate of growth of this function, however, is not much larger than a regular factorial.
Superfactorials
The superfactorial of n, written as n$ (the $ should really be a factorial sign ! with an S superimposed) has been defined as
- n$ = n(4)n
where the (4) notation denotes the hyper4 operator, or using Knuth’s up-arrow notation.
The sequence of superfactorials starts:
- 1$ = 1
- 2$ = 22 = 4
source: collections from books, internet etc
The factorial operation is encountered in many different areas of mathematics, notably in combinatorial, algebra and mathematical analysis. Its most basic occurrence is the fact that there are n! ways to arrange n distinct objects into a sequence.