# How 0!=1? Permutations and combinations

Usually n factorial is defined in the following way:

n! = 1 x 2 x 3 x … x n

A first way to see that 0! = 1 is by working backward. We know that:

1! = 1

2! = 1!*2

2! = 2

3! = 2!*3

3! = 6

4! = 3!*4

4! = 24

We can turn this around:

4! = 24

3! = 4!/4

3! = 6

2! = 3!/3

2! = 2

1! = 2!/2

1! = 1

0! = 1!/1

0! = 1

In this way a reasonable value for 0! can be found.

How can we fit 0! = 1 into a definition for n! ? Let’s rewrite the usual definition with recurrence:

1! = 1

n! = n*(n-1)! for n > 1

Now it is simple to change the definition to include 0! :

0! = 1

n! = n*(n-1)! for n > 0