Why is 0! equal to 1?

Why is 0! equal to 1?

If you're done with your permutations/combinations lecture, you must be wondering what is 0! And even if you know that it is 1, why is it so? 

This can be explained very simply using different approaches to combinations; I'll do it in three very basic ways : 

1. Using the formula: n! = n(n-1)!
2. Using a pattern 
3. The Basic Concept of Factorials. 

Proof 1:

We know that n! = n(n-1)(n-2)(n-3)(n-4).....(3)(2)(1)

Hence this can be written as n!= n(n-1)!

Lets use this fact to prove that 0! = 1 :

n!= n(n-1)!

Dividing both sides with n , we get: 

  

Now lets suppose that n=1:

we get:

 

Hence shown that 0! = 1

Proof 2:

Lets simply workout some factorials:

5! = (5)(4)(3)(2)(1)= 120

4!= (4)(3)(2)(1) = 24

3!= (3)(2)(1) = 6

2!= (2)(1) = 2

1!= 1

Do you notice a pattern? 

Note that 4! can be written as: 

Similarly, 

 

Following the same pattern, lets workout 0! : 

Hence both the explanations show that 0! = 1

Proof 3:

Lets just come back to the concept of factorials. What are factorials used for? 

Factorials help mathematicians calculate the possible combinations or permutations of something.

Lets explain it with an example:

If there are 3 parking spaces available, and there are also 3 cars to park, in how many different ways can the three cars be parked? The answer is 3! = 6 ways

Remove one parking space and one car from the scenario, if there are 2 parking spaces and 2 cars, how many different ways can they be parked? The answer is 2! = 2 ways

Remove another car, and parking space, how many different ways are available to park 1 car in the only parking space? There is just one way. Hence 1! = 1 way

NOW this is where it gets interesting. How many different ways are there to park no car in no parking space? Just one way. You don't. Yes, the ONE and only possibility is that you don't. This means that 0! = 1 way

HENCE PROVED THAT 0! = 1