The glib answer - it is needed for all of the math to work
The real answer - ooooookaaaayâŚ
My favorite way to think about factorial numbers in that n!
means âhow many different ways can I arrange n
balls of different colorsâ.
Suppose I have 3 balls, ['red', 'yellow', 'blue']
.
There are n! = 3 * 2 * 1 = 6
ways for me to arrange these balls:
['red', 'yellow', 'blue']
['red', 'blue', 'yellow']
['yellow', 'red', 'blue']
['yellow', 'blue', 'red']
['blue', 'red', 'yellow']
['blue', 'yellow', 'red']
From this you can see, this ball arranging definition makes sense, because I have 3
choices for my first ball. Once Iâve picked my first ball, I have 2
choices for my second ball, and after that I have 1
choice for my remaining ball. Thatâs a total of 6
different ball arrangements.
This pattern works for any number of balls.
Now, how many different ways can you arrange 0
balls?
âŚ
Exactly 1
way! You do nothing, and that is the only possible arrangement.
Looking back at the glib answer now, this âball arranging logicâ shows up in a lot of deep mathematical logic, and this logic only works out right if we consistently use the idea that there is only 1
way to arrange 0
balls (or the equivalent statement for the various deep mathematical logic statements under consideration).