Understanding the Concept of Factorial and the Reason Behind 0! = 1

Method 1 of 3 :-

Well there are many different ways to prove this:

One good way to prove this is by going back to the laws of exponents. (Long Proof):-

The one such law was :-

--> {x^0 = 1}

Going back on how this was derived, we can examine the periodicity of the series:-

Let say x = 2,

2^1 = 2

2^2 = 4

2^3 = 8

2^4 = 16

Now lets divide by 2 as we go upwards.

2^0 = 1

2^1 = 2   

2^2 = 4

               2^3 = 8

We get ( 2^0 = 1)

Similarly do this for 0! , 1!, 2! and for n!

As we go from down to up, divide by the respective n in the column.

1! = 1

2! = 2 ( 1 x 2)

3! = 6 (1 x 2 x 3)

Now divide by 3 , 2 , 1

0! = 1

1! = 1/1 = 1

2! = 2/2 = 1

3! = 6/3 = 2

Hence -> 0! = 1.

Method 2 of 3 :-

To arrange objects in every order possible, maximum possible are n!

(Pigeonhole principle)

For example:-

To arrange 3 objects :- order = 3! = 6 (ways possible)

Let say, there are 3 objects:- ABC

The 6 different orders are :- (For ABC are) :-

1. ABC

2. ACB

3. BAC

4. BCA

5. CAB

6. CBA

Similarly, to arrange 0 objects, there is only a (0) way to arrange them, that is 1 way.

Method 3 of 3 :-

The Gamma Function:-

The above function is equal to (z-1)!. After substituting z =0, we get:-

Gamma (0) = ∫ e^-t

_____ dt

t

We hence get the area covered by the integral = 1, when plotted.