FACTORIAL N: FACTS

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?

Christian Kramp. Kramp was born in Strasbourg, France in 1760. He used the notation n! in 1808 in one of his books, "Elements d'arithmétique Universelle". 

 Multiple scientists worked on this subject, but the principal inventors are J. Stirling in 1730 who    gives the asymptotic formula after some work in collaboration with De Moivre, then Euler in 1751 and finally C. Kramp and Arbogast who introduces between 1808 and 1816 the actual notation: n!. Of course other scientists such as Taylor also worked a lot with this notation.

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?

Triangular number. The first triangular number is 1. The second one is 2 + (2-1) = 2 + 1 = 3. The third triangular number is given by 3 + (3-1) + (3-2) = 3 + 2 + 1 = 6. This is followed by 10, 15, 21. Another simpler way to calculate the nth triangular number would be by using the formula [(n)(n+1)]/2. For instance, the 5th triangular number is [(5)(6)]/2 = 15. 

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?

5! - 1 = 119. 119 is the product of 2 smaller primes, which are 7 and 17. Some other factorial primes are such as 1! + 1 = 2, 2! + 1 = 3, 3! + 1 = 7.

 

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?

n!/(n-r)!. For permutation, we wish to find the number of different ways one can arrange r items in different orders, selected from a total of n items.

The factorial of 0, namely 0! is 0.

F. Actually, 0! = 1. We notice the followings: 1! = 1 

2!= 2 x 1 = 2 

3! = 3 x 2 x 1 = 6 

4! = 4 x 3 x 2 x 1 = 24 

5! = 5 x 4 x 3 x 2 x 1 = 120  Working backwards, we will get:

 4! = 5!/5 = 24 

 3! = 4!/4 = 6 

 2! = 3!/3 = 2

 1! = 2!/2 = 1 

 0! = 1!/1 = 1 

 Therefore, the operation of  C(5, 0)  is valid and possible, where the answer is 1. This means that we have only 1 way of choosing 0 items from a total of 5 items. 

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?

n!/(n-r)!. For permutation, we wish to find the number of different ways one can arrange r items in different orders, selected from a total of n items.

 

A positive integer that is in the form of n^{(n-1)^[(n-2)^...^{2^(1)}]} is called a/an ?

Exponential factorial. Let f(n) be the exponential factorial function for an integer n. When n = 1, f(n) = 1 When n = 2, f(n) = 2^1 = 2 When n = 3, f(n) = 3^(2^1) = 9 When n = 4, f(n) = 4^[3^(2^1)] = 262144 It should be noticed that when calculating the exponential factorial for n = 4, the operation involved is 4^[3^(2^1)], not [(4^3)^2]^1

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?

Taylor's theorem. Some of the famous power series that is derived from this Taylor's theorem is cos x = 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + (x^8)/8! -... Apart from this, the functions sin x, tan x, and e^x can also be expressed as power series. 

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

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