** What is the Factorial of Hundred**

Mathematical equations can be modeled on a scale of 1-10 with 10 being the highest level of difficulty. In this article, we will explore the factorial of numbers at different levels of difficulty starting from 100. Numbers that are set higher in difficulty have a smaller factorial that is evaluated by multiplying the number by the number of factors it has. * For example, *a 4 digit number with a 10 factor would be evaluated as 100 x 10 x 10 x 10 which equals 10000.

The factorial of a number is the product of all whole numbers from 1 to that number. The factorial of 100, for example, would be computed by multiplying 1*2*3*4*5… and so on until you get 100 terms. This is the equivalent of asking “What number multiplied by each of the numbers between 1 and itself equals 100? The factorial of 100 is the product of all positive integers which can be written as a product of numbers less than or equal to 100, including 0. There are 5 factorials of 100 that are possible with non-zero elements.**9.332622e+157 is the factorial of hundred.**

**व्हाट इस थे फ़ैक्टोरियल ऑफ़ १००**

*The Factorial of hundred* is 9.332622e+157

**How do you solve 100 factorial?**

To solve 100 factorial, you would start by writing 100 as the base number and then multiplying it by 99. You would continue this process, multiplying the previous result by 98, 97, 96 and so on until you reach 1. The answer would be the final result of this calculation.

For those who are not familiar with factorials, they are basically a mathematical way to represent repeating multiplication. So, 100 factorial would be written as 100! And it would equal to 100x99x98x97x96x95… all the way down to 1.

**What is a factorial of 0?**

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: 5! = 5 × 4 × 3 × 2 × 1 = 120. The value of 0! is 1, according to the convention for an empty product.

**Why is 0 != 1 prove it?**

The factorial of 0 is 1, because there are no positive integers less than 0.The factorial of 1 is 1 too, because the only positive integer less than 1 is 1. It’s easy to see that the factorial of any positive integer n must be greater than or equal to n. In other words, 0! = 1.

**What is Factorial?**

Factorials are a type of number with a special property – multiplication.

The factorial of a particular whole number is the product of all the whole numbers less than or equal to that number, raised to each of those numbers raised to the power one fewer.* For example,* 3!=3x2x1=6 and 10!=10x9x8x7x6x5x4x3x2x1 =3.Factorials are common mathematical terms that are used in everyday conversations. A factorial of two is an even number, while a factorial of three is an odd number. Factorials are always multiplied by the number that precedes them. Without multiplying, they would be written as 1×2=2, 2×3=6, 3×4=12, and so on. While this may seem simple enough to some, others may find this difficult to understand.

There are many mathematical concepts that can be difficult to understand, but this is especially true for the definition of “factorial.” The word is often explained in terms of an equation, but it can also be described in simpler language. A factorial number is a product of multiplying consecutive integers together. For example, if 1x2x3x4=24, then 4! = 24 because 1x2x3x4 = 6 x 6 x 6 x 6.

* Mathematically, *factorial is a simple concept. Factorials are simply products. Factorials are indicated by an exclamation point. Factorial is the multiplication of all the natural numbers that are less than it by all the natural numbers.Any number can be found by using the factorial formula. In other words, it’s the product of the number and all the lower value numbers until 1. This is the result of multiplying the descending series of numbers. One must remember that the factorial of 0 is 1. Permutations and combinations of the Factorial Formula are used in probability calculation in many direct and indirect ways. Factorial functions include double factorial, multifactorial, etc. Gamma is also an important concept based on factorial. Factorial of a number is the function that multiplies the number by every natural number below it. It can be symbolized as “!”. Thus, n factorial represents the product of the first n natural numbers!

n! or “n factorial” means: n! = 1. 2. 3…where, n = Product of the first n positive integers = n(n-1)(n-2)…………………….(3)(2)(1)

**How Factorial is calculated?**

The factorial of a number is calculated by using formula:

n! = n × (n-1) × (n-2) × (n-3) × …× 3 × 2 × 1

The recurrence relation of a factorial number is the product of the factorial number and the factorial of that number less 1. It is expressed as follows:

n! = n. (n-1)!

**History of Factorials**

Since the late 15th century, factorials have been studied by western mathematicians. Italian mathematician Luca Pacioli calculated factorials up to 11! , in a 1494 treatise in connection with a problem of dining table arrangement. A 1603 commentary by Christopher Clavius on the work of Johannes de Sacrobosco discussed factorials, and a 1640 article by French polymath Marin Mersenne published large (but not entirely correct) tables of factorials, up to 64!, based on the work of calvivus.

It was Isaac Newton who first formulated, in a letter to Gottfried Wilhelm Leibniz, the power series for the exponential function with its reciprocals for its coefficients in 1676. There are also important early works in European mathematics on factorials, including an extensive discussion in John Wallis’s 1685 treatise, Abraham de Moivre’s study of their approximate values in 1721, and a 1729 letter from James Stirling to de Moivre formulating the continuous extension of the factorial function to the gamma function. Legendre’s formula, which describes the exponents in factorizing factorials into prime powers, was included in an 1808 text on number theory by Adrien-Marie Legendre.

The French mathematician Christian Kramp introduced the factorial notation in 1808. It has been replaced by many other notations ever since. For a time in Britain and America, another notation, which enclosed the factorial argument with left and right sides of a box, was popular, but fell out of use, perhaps because it is difficult to typeset. Factorial (originally French: factorielle) was used for the first time in 1800 by Louis François Antoine Arbogast in the first work on Faà di Bruno’s formula, but it referred to a more general concept of products of arithmetic progressions. The “factors” that this name refers to are the terms in the formula for the product.

**List of factorial values from 1 to 25**

n | Factorial of a Number n! | Value |

1 | Factorial of 1 (1!) | 1 |

2 | Factorial of 2 (2!) | 2 |

3 | Factorial of 3 (3!) | 6 |

4 | Factorial of 4 (4!) | 24 |

5 | Factorial of 5 (5!) | 120 |

6 | Factorial of 6 (6!) | 720 |

7 | Factorial of 7 (7!) | 5,040 |

8 | Factorial of 8 (8!) | 40,320 |

9 | Factorial of 9 (9!) | 362,880 |

10 | Factorial of 10 (10!) | 3,628,800 |

11 | Factorial of 11 (11!) | 39916800 |

12 | Factorial of 12 (12!) | 479001600 |

13 | Factorial of 13 (13!) | 6227020800 |

14 | Factorial of 14 (14!) | 87178291200 |

15 | Factorial of 15 (15!) | 1.3076744e+12 |

16 | Factorial of 16 (16!) | 2.092279e+13 |

17 | Factorial of 17 (17!) | 3.5568743e+14 |

18 | Factorial of 18 (18!) | 6.4023737e+15 |

19 | Factorial of 19 (19!) | 1.216451e+17 |

20 | Factorial of 20 (20!) | 2.432902e+18 |

21 | Factorial of 21 (21!) | 5.1090942e+19 |

22 | Factorial of 22 (22!) | 1.1240007e+21 |

23 | Factorial of 23 (23!) | 2.5852017e+22 |

24 | Factorial of 24 (24!) | 6.204484e+23 |

25 | Factorial of 25 (25!) | 1.551121e+25 |

**Factorial for Negative Number**

We cannot have factorials for negative numbers like −1, −2, and other negative integers because negative integer factorials are undefined.

**Factorial of 0 (Zero)**

It is well known that the factorial of 0 is equal to 1 (one). It can be represented as: **0! = 1**

The notation and definition stipulated above are based on multiple reasons. As a start, the definition allows for the compact expression of a number of formulae, including the exponential function, and it creates an extension of the recurrence relation to zero. * Furthermore*, where n = 0, the definition of its factorial (n!) encompasses the product of no numbers, which is equivalent to the multiplicative identity. In addition, the definition of the zero factorial only includes one permutation of zero or no objects. A number of identities in combinatorics are also validated by the definition.

**Why there is no factorial for negative numbers?**

Negative numbers do not have a factorial because they cannot be used to represent a quantity of objects arranged in a line, as the factorial definition requires. Additionally, the factorial function is not defined for negative values. This is because when you multiply a negative number by any other number, the product is always positive. For these reasons, it does not make sense to talk about the factorial of a negative number.

**What is infinity factorial?**

In mathematics, the infinity factorial is the infinite product of all natural numbers. That is, infinity factorial is equal to:

infinity! = ∞ × (∞ – 1) × (∞ – 2) × ⋯

The infinity factorial can be used to calculate the number of ways that a given number can be partitioned into a sum of positive integers. For example, if we want to find the number of ways that 5 can be partitioned, we would calculate:

5! = 5 × 4 × 3 × 2 × 1 = 120

The answer tells us that there are 120 different ways to partition 5.

**Applications of Factorial Value**

There is the use of factorial in mathematics. The following are some applications of factorial in mathematics:

**Recursion**

We can use factorial in the recursive definition of a number. Numbers may be expressed as expressions containing only the number. It is represented as:

**p!=p×(p–1)×(p–2)×(p–3)..(p−(p–2))×(p–(p–1))**

**Permutations**

Permutations are the arrangement of given r things out of total n things when strict order is required.Basically, a permutation is an arrangement of things from a set where the order in which the things are selected matters. Accordingly, we can say that every language consists of some particular permutations of a collection of alphabets put together. It is represented by:

**Pnr=n!(n−r)!**

**Combinations**

Combinations is an arrangement of n things out of r things when order is not important. It is represented by:

Cnr =n!(n−r)!r!

**Probability Distributions**

Numerous probability distributions include factorials, including binomial distributions. Permutations and combinations are often used to find the probability of an event.

**Number Theory**

In number theory, as well as in approximation, factor values are used extensively.

**Examples of Factorial Formula**

**Find the value of 8!.**

Solution:

The formula for factorial is,

p! = p×(p−1)×(p−2)×(p−3)…×2×1

8! = 8×(8−1)×(8−2)…×2×1

= 8×7×6×….3×2×1

= 40320

**Find the factorial of 0.**

Solution:

The factorial of 0 is 1

i.e., 0 ! = 1

The empty product convention states that the result of multiplying no factors is the nullary product. This indicates that the convention is equal to multiplicative identity.

**Sequences related to Factorial**

There are several sequences which are similar to or related to the factorials:

**Double Factorials**: The factorials which are used to simplify trigonometric integrals is known double factorials.**Multi-factorials**: The factorials which can be denoted with multiple exclamation points is known as multi-factorials.**Primorials**: The factorials which entail getting the product of the prime numbers, which are less than or equal to*n*is known as primorials.**Super-factorials**: The factorials which are defined as the product of the first*n*factorials are super factorials.**Hyper-factorials**: The factorials which are a result of multiplying a number of consecutive values ranging from 1 to*n*are hyper factorials.

**Queries related to Factorials**

**How can we calculate factorials?**

**Ans:** Multiply the given number with the previous number’s factorial value to find the factorial of the given number. Let’s look at the value of 6. By multiplying 120 by 6, we get 720.

**What are factorials used for?**

**Ans:** Factorial functions are used for calculating combinations and permutations.

**What does factorial rule mean in statistics?**

**Ans:** The factorial is a mathematical operation in which you multiply a whole number by all the whole numbers below it. In other words. **= n × ( n − 1 ) × …** **× 2 × 1** .

**In what way does factorial represented?**

Ans: A factorial is a mathematical formula represented by an exclamation mark “!”. For instance, the factorial of eight is represented as 8!

**Permutation and Combination**

The methods used to count how many possible outcomes there are in different circumstances are permutation and combination. Permutations are considered arrangements, and combinations are considered selections. According to the fundamental principle of counting, there are sum rules and product rules. The concept of factorials must be recalled in order to comprehend permutation and combination. The product of the first n natural numbers is n! and the number of ways of arranging n unlike objects is n!.

**Permutation**: Permutations are arranged in a definite order of a number of objects taken all or some at once. Let’s take the following 10 numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. The number of different 4-digit-PINs that can be derived using these 10 numbers is 5040. P(10,4) = 5040. The permutations of four numbers taken from a set of ten numbers are equal to the factorial of ten divided by the factorial of the difference between ten and four. It is easily calculated by using nP_{r}= n!/(n-r)!**Combinations:**Combinations are all about grouping. We can calculate the number of possible groups from the things available using combinations. The combination of ‘r’ persons from the available ‘n’ persons is given as nC_{r}= n!/r!(n-r)!.

**Permutation Vs Combination**

It is necessary to understand the difference between the permutation and combination in order to know when to apply each. Permutation consists of all the different arrangements that can be made when things of different kinds are put together. Combination describes the ability of smaller groups or sets to be derived from a larger set. The combination of things is all that matters, and the arrangement of the individual elements within the group is not important. We can better understand the difference between permutation and combination by looking at the below table.

Permutation | Combination |
---|---|

A permutation is a way of arranging a set of objects in a sequential order. | Combination involves several ways of selecting objects from a large pool, such that their order is irrelevant. |

The order in which the elements are placed is very important. | The order in which the elements are selected is not important. |

Permutation can be made using repetition or without repetition of elements. | Combination is not concerned with repetition or without repetition of elements. |

Permutation is all about arranging people, digits, numbers, alphabets, letters, and colours. | Combination is all about selection of menu, food, clothes, subjects, team. |

Picking a team captain, pitcher and shortstop from a group. | Picking three team members from a group. |

Picking two favourite colours, in order, from a colour brochure. | Picking two colours from a colour brochure. |

Picking first, second and third place winners. | Picking three winners. |

**Formulas of Permutation and Combination**

We analyze the counting situation to determine whether to utilize combination or permutation. Accordingly, we apply the permutation and combination formulas.

**Formula 1:**The formula for calculating factorial of a natural number n is:

**n! = 1 × 2 × 3 × 4 × …….× n**

**Formula 2:**The number of distinct permutations of r objects that can be made from n distinct objects is represented as:

** nPr = , where 0 ≤ r ≤ n.**

**Formula 3:**The number of permutations of n different things, taken one at a time, if repetition is allowed, is n^{r.}**Formula 4:**Permutations of n objects taken all at once, with objects are of the first kind, . Similarly, there are objects are of the second kind, …, objects are of the k^{th}kind and the rest if any, and if all are different is**Formula 5:**There are n combinations of different objects taken r at a time, so the number of combinations is

**nCr = , where 0 ≤ r ≤ n.**

In some cases, this formula is also referred to as the ncr formula.

**Formula 6:**A relationship between permutations and combinations for r things taken from n things.

**nP _{r}= r!× nC_{r}**

**Derivation of Permutation Formula**

Let us assume that there are r boxes and each of them can hold one thing. There will be as many permutations as there are ways of filling in *r *vacant boxes by *n* objects.

- No. of ways the first box can be filled:
*n* - No. of ways the second box can be filled:
**(***n*– 1) - No. of ways the third box can be filled:
**(***n*– 2) - No. of ways the fourth box can be filled:
**(***n*– 3) - No. of ways r
^{th}box can be filled:**[***n*– (*r*– 1)]

The number of permutations of *n *differe0nt objects taken *r *at a time, where 0 < *r ≤ n *and the objects do not repeat is: *n*(*n* – 1)(*n* – 2)(*n* – 3) . . . (*n* – r + 1)

nP_{r} = n(n – 1)(n – 2)(n – 3). . .(n – r + 1)

Multiplying and dividing by (*n *– *r*) (*n *– *r *– 1) . . . 3 × 2 × 1, we get:

**Derivation of Combination Formula**

Let us assume that there are r boxes and each of them can hold one thing.

- No. of ways to select the first object from
*n*distinct objects:*n* - No. of ways to select the second object from (
*n-1)*distinct objects:**(***n-1)* - No. of ways to select the third object from (
*n-2)*distinct objects:**(***n-2)* - No. of ways to select r
^{th}object from [*n-(r-1)*] distinct objects:**[***n-(r-1)*]

Completing the selection of r things from the original set of n things creates an ordered subset of ** r** elements.

The number of ways to make a selection of r elements of the original set of

*n*elements is:

**.**

*n*(*n*– 1) (*n*– 2) (n-3) . . . (*n*– (*r*– 1)) or*n*(*n*– 1) (*n*– 2) … (*n*–*r*+ 1)*Let us consider the ordered subset of r elements and all its permutations. The total number of all permutations of this subset is equal to r! because r objects in every combination can be rearranged in r! ways.*

**Hence,** the total number of permutations of *n *different things taken *r *at a time is (nC_{r}×r!). It is nothing but nP_{r}.

**nP _{r = }nC_{r * }r!**

**nC _{r }= nP_{r }/ r! = n!/(n-r)! r! **

**References**

*Byjus*. (n.d.). Retrieved from Byjus: https://byjus.com/maths/factorial/

*Corporate Finance Institute*. (n.d.). Retrieved from Corporate Finance Institue: https://corporatefinanceinstitute.com/resources/knowledge/other/factorial/

*Cuemath*. (n.d.). Retrieved from Cuemath: https://www.cuemath.com/data/permutations-and-combinations/

*Cuemath*. (n.d.). Retrieved from Cuemath: https://www.cuemath.com/numbers/factorial/

*embibe*. (n.d.). Retrieved from embibe: https://www.embibe.com/exams/permutation-and-combination/

*toppr*. (n.d.). Retrieved from toppr: https://www.toppr.com/guides/maths-formulas/factorial-formula/#:~:text=The%20factorial%20of%20a%20number,theory%2C%20approximations%2C%20and%20statistics.

*Wikipedia*. (n.d.). Retrieved from Wikipedia: https://en.wikipedia.org/wiki/Factorial

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