Everything You Need to Know About Factorials

In mathematics, a factorial is the product of all non-negative integers that are less than or equal to a non-negative integer (n) that is given. The symbol n (positive integer) is ended with the symbol of factorial – an exclamation (!). The fundamental representation of factorial is n! = n (n-1)(n-2)……2(1). Here, n! represents the multiplicative product of all the non-negative integers lower than or equal to the integer n. The factorial was first defined by an author from Britain named Fabia Stedman in 1677. He defined a factorial as an equivalent to change ringing. Change ringing is a sequence in a musical performance that involves ringing multiple tuned bells. Later, in 1808, a French mathematician established a symbol for factorials – n!. In mathematics, factorial is important in various areas like probability, graph theory, statistics, algebra, number theory, and many more. 

What is factorial? 

This section will address the question – what is factorial?

The factorial of a number (n) is defined as a function that multiplies all the non-negative numbers less than n. 

Representation for n factorial – n!=(n-1)(n-2)……..(2)(1). 

Here, n is a positive integer. 

Let us look at an example – 5! =5 4 3 2 1 = 120

What is zero factorial? 

The factorial of zero is equivalent to 1. It is represented as 0! = 1 

For n = 0, n! involves products of no numbers. Hence, this is an example of an empty product (no factors for a product) equivalent to the multiplicative identity in broader terms. 

The zero factorial definition involves only a single permutation of zero (no objects). This definition also justifies various identities in combinatorics. 

Terms to be familiar with when defining zero factorial. 

• Combinatorics – this field of mathematics majorly focuses on counting
• Permutation refers to the arrangement of objects of a set into a sequence

Factorial of negative numbers

This section will see if we can have factorial values for negative numbers like -2, -4, etc. 

Let us start from 4 factorial and go downwards until we reach a negative value of -1.4! = 4 3 2 1 = 24.

3! =4!/4 =6.

2! = 3!/3 =2.

1! =2!/2.

0! = 1!/ 1 = 1.

-1! = 0!/0 = 1/0.

The value for the negative number -1 is undefined, as 1 cannot be divided by zero. From this, we can conclude that factorial of negative numbers is not possible. 

Uses of factorial 

Permutations 

One of the primary uses of factorial is the area of permutation. There are n! Variety of ways of arranging n distinct items. The formula for permutation is