Factorial and why 0! = 1.

We all are familiar with factorials and their implications in combinatorics, algebra and mathematical analysis and we all have seen this identity or say result 0!=1, today we are going to learn why this is true both mathematically and intuitively. For those who are new to this concept here is the definition of factorial

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,

In general, n factorial is given by :

Now, first we will work out some mathematics to prove 0! = 1.

Completing the pattern :

The best way to solve a problem sometimes is to find the pattern using some values and let's do the same we will start from 6! and go down to 0!, here is the figure :

Now we can see that 5! = 6!/6, 4! = 5!/5, 3! = 4!/4, 2! = 3!/3, 1! = 2!/2 and so in order to complete the pattern 0! must be equal to 1!/1.

Another way of expressing the above pattern is an equation :

Putting the value n=1, we get 0! = 1!/1 hence the value of 0! is 1.

Now, that was the mathematical proof, now for the intuitive way we need to know what is the significance of n! in real world.

Factorial n is the number of ways to arrange n objects or we can say to shuffle n objects.

For example 4 objects can be shuffled in 24 ways as shown :

Similarly 3 objects can be arranged in 6 ways :

Now, 2 objects can be arranged in 2 ways and 1 object can only be arranged in one way, but what about null object, null object being no object or absence of an object just like a set has a null element, just because it is null we do not ignore this element and that's how we define things in mathematics, the absence of something is is still something which counts one time hence we can say the ways in which we can arrange 0 objects is 1 way hence

0! =1

So, that was another way to analyse the same thing, hope you guys liked the article, have a nice day!