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## #1 2021-12-13 17:17:41

ganesh Registered: 2005-06-28
Posts: 38,003

### Factorial

Factorial

Gist

factorial, in mathematics, the product of all positive integers less than or equal to a given positive integer and denoted by that integer and an exclamation point. Thus, factorial seven is written 7!, meaning 1 × 2 × 3 × 4 × 5 × 6 × 7. Factorial zero is defined as equal to 1.

Factorials are commonly encountered in the evaluation of permutations and combinations and in the coefficients of terms of binomial expansions (see binomial theorem). Factorials have been generalized to include nonintegral values (see gamma function).

Details

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n:

For example,

The value of 0! is 1, according to the convention for an empty product.

The factorial operation is encountered in many areas of mathematics, notably in combinatorics, algebra, and mathematical analysis. Its most basic use counts the possible distinct sequences – the permutations – of n distinct objects: there are n!.

The factorial function can also be extended to non-integer arguments while retaining its most important properties by defining x! = Γ(x + 1), where Γ is the gamma function; this is undefined when x is a negative integer.

History

The use of factorials is documented since the Talmudic period (200 to 500 CE), one of the earliest examples being the Hebrew Book of Creation Sefer Yetzirah which lists factorials (up to 7!) as a means of counting permutations. Indian scholars have been using factorial formulas since at least the 12th century. Siddhānta Shiromani by Bhāskara II (c. 1114–1185) mentioned factorials for permutations in Volume I, the Līlāvatī.

In the 1640s, French polymath Marin Mersenne published large (but not entirely correct) tables of factorials, up to 64!. In 1677, Fabian Stedman later described factorials as applied to change ringing, a musical art involving the ringing of several tuned bells. After describing a recursive approach, Stedman gives a statement of a factorial (using the language of the original):

Now the nature of these methods is such, that the changes on one number comprehends [includes] the changes on all lesser numbers ... insomuch that a compleat Peal of changes on one number seemeth to be formed by uniting of the compleat Peals on all lesser numbers into one entire body.

The notation n! for factorials was introduced by the French mathematician Christian Kramp in 1808.

Definition

The factorial function is defined by the product

for integer n ≥ 1. This may be written in pi product notation as

This leads to the recurrence relation

For example,

and so on.

Factorial of zero

The factorial of 0 is 1, or in symbols, 0! = 1.

Factorial of a whole number 'n' is defined as the product of that number with every whole number till 1. For example, the factorial of 4 is 4×3×2×1, which is equal to 24. It is represented using the symbol '!' So, 24 is the value of 4!  In the year 1677, Fabian Stedman, a British author, defined factorial as an equivalent of change ringing. Change ringing was a part of the musical performance where the musicians would ring multiple tuned bells. And it was in the year 1808, when a mathematician from France, Christian Kramp, came up with the symbol for factorial: n! The study of factorials is at the root of several topics in mathematics, such as the number theory, algebra, geometry, probability, statistics, graph theory, and discrete mathematics, etc.

What Is Factorial?

The factorial of a number is the function that multiplies the number by every natural number below it. Symbolically, factorial can be represented as "!". So, n factorial is the product of the first n natural numbers and is represented as n!

n factorial

So n! or "n factorial" means: n! = 1. 2. 3…………………………………n =  Product of the first n positive integers =  n(n-1)(n-2)…………………….(3)(2)(1)

For example, 4 factorial, that is, 4! can be written as: 4! = 4 × 3 × 2 × 1 = 24.

Observe the numbers and their factorial values given in the following table. To find the factorial of a number, multiply the number with the factorial value of the previous number. For example, to know the value of 6! multiply 120 (the factorial of 5) by 6, and get 720. For 7! multiply 720 (the factorial value of 6) by 7, to get 5040.

n  :   n!
1  :  1
2  :      2 × 1 = 2
3  :  3 × 2 × 1 = 3 × 2!  =  6
4  :  4 × 3 × 2 × 1  = 4 × 3!  =  24
5  :  5 × 4 × 3 × 2 × 1  = 5 × 4!  = 120

Formula for n Factorial

The formula for n factorial is: n = n × (n − 1)!

The factorial of a number has many and intensive uses in permutations, combinations and the computation of probability. We represent it by an exclamation mark (!). Factorials are also used in number theory, approximations, and statistics. In this topic, we will discuss the Factorial Formula with examples. We shall also learn the various applications of factorial formula such as permutations, combinations, probability distribution, etc. Let us start!

Definition of Factorial

The factorial formula is used to find the factorial of any number. It is defined as the product of the number with all its successive lowest value numbers till 1. Thus it is the result of multiplying the descending series of numbers. It must be remembered that the factorial of 0 is 1. Factorial Formula has many direct and indirect applications in permutations and combinations for probability calculation.

There are various functions based on factorials like double factorial, multifactorial, etc. Also, the Gamma function is an important concept based on factorial.

Factorial Formula

Formula for the Factorial : To get the factorial of a given number n the following given formula can be used,

…....

This is possible due to the recursive nature of factorial computation.

Let us understand it with some examples.

Some Applications of Factorial Value:

Some applications of factorial in mathematics are as follows:

1) Recursion

In the recursive definition of a number, we may use factorial. A number can be expressed in an expression containing the number only.

......
.

2) Permutations

Arrangement of given r things out of total n things when order is strictly important.

3) Combinations

Arrangement of given r things out of total n things when order is not important.

4) Probability Distributions

There are various probability distributions like binomial distribution which include the use of factorial. To find the probability of an event, the concept of permutations and combinations is used a lot.

5) Number Theory

Factorials value are used extensively in number theory and also for approximations.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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