Why does the 0! Factorial is 1?

Submitted by Aniket Dutta (Department of BCA (Batch : 2017-2020))

We all about know that ,the value of factorial zero(0!) is 1,but we didn’t khow the reason why does the value of zero factorial (0!) is 1.

 Let us the discuss  about it………..

We have a whole number ‘n’, then-------

n!=n.(n-1).(n-2).(n-3)……………3.2.1.------------(1)

let take ,n=5…so,we get the above equation….

5!=5*4*3*2*1=120.

Now,we have the following equation…..

4!=5!/5=24 ; 3!=4!/4=6 ; 2!=3!/3=2 ; 1!=2!/2=1;

and  0!=1!/1=1;

S0, according to the equation (1) we easily find out to the value of zero factorial (0!) is 1.

But that is not enough reason……..

Let check the another reason for find out the value of zero factorial…

Using Permutation------------

 For example, there are given a set {1,2,3},

which contain three element.

Since,we may write this elements in following six ways……….

{1,2,3} , {1,3,2} , {2,3,1} , {2,1,3} , {3,1,2} , {3,2,1}.

So,we could also state this fact through the equation 3!=6,which is a factorial representation of the full set of permutations.

Similarly,there are 4!=24 permutations of a set            with four elements and 5!=120 permuations of a set with five elements.

So, we think about factorial is to let ‘n’ be a natural number and say that n! is the permutation for a set with n elements.

       Hence, a set with one element has a single permutation, as the element 1 in the set {1} can only be ordered in one way. But the set with zero element is called empty set. So,we ask how many  ways can we order a set with no element ,hence there is nothing to put it in an order ,thus we have that 0!=1.

Using Combination………………

     For example, consider the set {1,2,3} where in there is one combination consists of all three element.

We use the formula for combination with the combination of three elements taken three at a time and see that………

C(3,3)=1               using C(n,r)=n!/r!*(n-r)!.

   =>3!/3!*0!=1

   =>3!*0!=3!

   So, 0!=1.

There are other reasons why the definition of 0! = 1 is correct, but the reasons above are the most straightforward. The overall idea in mathematics is when new ideas and definitions are constructed, they remain consistent with other mathematics, and this is exactly what we see in the definition of zero factorial is equal to one.