why 0! is 1?

Answer 1

There is exactly one way NOT to arrange anything.

If n! is defined as the product of all positive integers from 1 to n, then:

1! = 1*1 = 1
2! = 1*2 = 2
3! = 1*2*3 = 6
4! = 1*2*3*4 = 24
...
n! = 1*2*3*...*(n-2)*(n-1)*n
and so on.

Logically, n! can also be expressed n*(n-1)! .

Therefore, at n=1, using n! = n*(n-1)!

1! = 1*0!

which simplifies to 1 = 0!

Answer 2

It is a definition that is used because it simplifies many formulas. The first answer hints at that, but remember it is nothing that can be proved--it is just defined that way for convenience.