How will you prove that 0! =1?


So it all started while I was attending my professor's math lectures in college one fine day. The topic was Permutations and Combinations and the basics start from knowledge about factorials.So we already know 1!=1, 2!=2, 3!=6 and so on... said my professor. Then he asked then what is 0! and does it really exists or not??

We all answered in excitement yes it exists and is equal to 1.

Then can you prove it. He argued. We were speechless. I told that its a convention adopted as per the definition of factorial that is defined over set of natural numbers. He was not quite impressed with my response.

In a flash of memory I remembered some properties of factorials.

n!=n*(n-1)!

I simply tried assuming the LHS of the expression as shown below

LHS = n! - (n-1)! and Substitute n=1. LHS becomes 1!-0! 

Save that . I simultaneosuly tried simplifying LHS to obtain RHS and substituted again n=1

RHS= (n-1)*(n-1)!                            

RHS=0 

and  we know that

LHS=RHS

1!- 0!=0

which means 0! =1

Hence proved and after that my professor was quite overwhelmed with this response :)

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