Why is the factorial of 0 equal to 1?

Question

Wouldn’t this also mean that 1!=0!, why is this true?

In: 383

Answer 1

Because “doing nothing” always count as one way to arrange stuff.

If I give you three objects [1,2,3], you can:

1. Do nothing -> [1,2,3]
2. Swap the first two -> [2,1,3]
3. Swap the last two -> [1,3,2]
4. Swap the first and last -> [3,2,1]
5. Take the last and add it before the first -> [3,1,2]
6. Take the first and add it after the last -> [2,3,1]

That’s 6 ways to arrange the three objects, so 3! = 6.

And as you see, the first way was “do nothing”. And you can always “do nothing”. Even if I give you zero object, you can “do nothing”. So when you count the arrangements, you will always get at least 1, corresponding to that “do nothing”. So 0! must be at least 1.

And obviously, if I give you zero objects, there is no other things to do than “do nothing”, so having 0! greater than 1 would be absurd. So 0! = 1.