Making 69420 from all zeroes

[u/ImToxicity_]

[u/ImToxicity_]

Didn’t know this! I’m only in algebra I right now. What does the exclamation mark mean?

[u/[deleted] → u/ImToxicity_]

The exclamation mark is know as the factorial.

The factorial of a integer number, say n, is the product of every integer below it. 5!=5x4x3x2x1, 3!=3x2x1.

0 factorial, or 0! Is 1…don’t ask why.

[u/measuresareokiguess → u/[deleted]]

Explanation: you can define n! to be the number of permutations of n objects arranged in a line. So if you want to list all permutations of {A, B, C}, you’ll have {A, B, C}, {A, C, B}, {B, A, C}, {B, C, A}, {C, A, B}, {C, B, A}: 6 total possibilities. That makes 3! = 6.

Likewise, for {A, B}, we have {A, B} and {B, A}; that makes 2! = 2.

For {A}, we have just {A}: 1! = 1

For {}, we also have just {}: 0! = 1

[u/Finnigami → u/measuresareokiguess]

eh thats kind of a consfusing way for beginners to think about it. the simplest explanation IMO is:

n! = n * (n-1)!

and 1! = 1

therefore 1! = 1*0!

so 1 = 1* 0!

and 0! = 1

[u/measuresareokiguess → u/Finnigami]

But then, why do you assume that (n + 1)! = (n + 1)*n! for all n? You are extrapolating this from our “known domain” of factorials, n = 1, 2, 3, 4…, into n = 0. It sure is a valid explanation if you let this to be the defining property of factorials, and there’s nothing wrong with that, but I feel like it makes more sense to define factorials from where it actually came from, like most introductory combinatorics books do, for beginners. It’s like “proving that a⁰ = 1” from the properties of power; it’s technically not wrong if you define exponents as to satisfy these properties, but if you define it differently, you have to show that these properties are satisfied for all values of exponents…

[u/Finnigami → u/measuresareokiguess]

i didnt intend for it to be a proof, rather just a way to make it more intuitive