Factorial

The Factorial is defined as the product of a number. The Factorial of a number is a function that multiplies the given number with every natural number being less than the given number. In other words, it is a product of positive integers that are less than or equal to the given number. The “exclamation” mark denotes it. It is an operation of multiplication of natural numbers which are equal to or less than the given number. It is all about the product of consecutive integers which are equal to less than the given integer with an exclamation mark.

Factorial

In Mathematics, Factorial is an important function. It is used to find out the number of ways the set of numbers can be arranged. Daniel Bernoulli discovered the well-known function of the Factorial.

In 1808, the French mathematician Christian Kramp introduced the symbol of the factorial function. The word “factorial” comes from the French word factorially. It shows the concept of products of arithmetic progression. The word factor denoted the terms of the product, which resulted in the Factorial by the multiplication of the factors of the given number.

The formula for n Factorial

The Factorial of a natural number “n” is the multiplication of all positive natural numbers equal to or less than “n” to generate a number which is the Factorial of “n.” It is represented as “n!”.

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

For an integer n ≥ 1, the general representation in terms of “π “product notation is,

            n! = i=1ni

It means that the Factorial of a number is the multiplication of all the numbers less than or equal to the number 1.

Factorial of First Five Numbers

The Factorial of the first five numbers from 1 to 5 is given below:

• 1! = 1 × (1-1) = 1 × 1 = 1
• 2! = 2 × (2-1) = 2 × 1 = 2
• 3! = 3 × (3-1) × (3-2) = 3 × 2 × 1 = 6
• 4! = 4 × (4-1) × (4-2) × (4-3) = 4×3×2×1 = 24
• 5! = 5 × (5-1) × (5-2) × (5-3) × (5-4) = 5 × 4 × 3 × 2 ×1 = 120

What Is 0!

The Factorial of the number “0” will always be 1.

               0! = 1

According to the convention of empty product, the Factorial of zero is null, which is equal to the multiplication of the identity number. 

What is 100 Factorial 

Let’s understand what 100 Factorial is. 

The 100 Factorial is the multiplication of all the natural numbers which are equal to or less than 100. The general representation is given as 100 Factorial. The calculation of 100 Factorial It is given below.

100 Factorial = i=1100100

100 Factorial = 100 × (100-1) × (100-2) × ……× 3 × 2 × 1

100 Factorial = 9.332621544 × 10157

Usage of Factorial 

The factorial concept is used in many mathematical concepts that are given below:

Permutation

Permutation represents the ordered arrangement of outcomes of the given set of numbers. The permutation is calculated by using the function of Factorial.

              nPr = n!/n-r! 

Combination

The combination represents a grouping of outcomes of the given set of numbers. The combination is calculated by using the function of Factorial.

nCr = n!/r!(n-r!) 

Properties of Factorial 

The significant properties of Factorial are as follows:

• Divisibility
• Computation
• Growth 

Divisibility

The function of Factorial has a property of divisibility. The Factorial of a number is divided by all positive integers up to the given number. The divisibility property of the function shows that the result of Factorial of all the even and odd numbers that are equal to or greater than 2 is always an even number. 

Computation

All scientific calculators and computer devices have one most common feature: the symbol of the factorial “!”. All the necessary computation takes place on scientific calculators and computers and other electronic computing devices by the symbol of “!”.

Growth

The Factorial is a function of the multiplication, which can be converted into the function of a sum by taking the natural logarithm of the Factorial, which shows the exponential growth of the given number.

Calculation of Factorial

The Factorial of n is denoted by n! and calculated by multiplying numbers from 1 to n. If the Factorial of (n-1) is given, then the value of Factorial of n can be calculated by multiplying the number n with Factorial of (n-1).  

n! = n × (n-1)!

For example, the value of 5! is 120. Then the value of 6! It is calculated as given below: 

    5! = 120

    6! = 6 × 5! = 6 × 120 = 720

Conclusion

The Factorial function was developed in the late 18th and early 19th centuries. In mathematics, many concepts related to number sequence, binomial coefficients, double factors, falling factorials, and sub-factorials are related to the function of Factorial. It is implemented in different computing styles. It is the most important feature in the software libraries of scientific computation. The factorial function is a concept of products of the arithmetic progression of the given number. It is the function of factors of the number which are in multiplication with each other.