Question

Prove that 0! = 1

Answer 1

It is defined that way. It makes possible the definition that

n! = n(n-1)! for all integer n>0

Answer 2

Suppose we have n objects.
In how many ways can I take all of them
of course the answer to that is 1 way.

but from the theory of combinations it would be
n!/( (n-n)(n!) )

= n!/(0!n!)
= 1/0!
but we know that has to be 1

1/0! = 1, this can only be true if we define 0! = 1

That is , we are forced to accept 0! = 1 if our theory of combinations and permutations is accepted.

Answer 3

Thanks for the help !