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n!/n = (n-1)!
1!/1 = 0! = 1
2! = 3^0 * 2* 1
1! = 3^0 * 2^0 * 1
0! = 3^0 * 2^0 * 1^0
I could include more natural numbers, all raised to the 0 power, but this is the basic idea behind an “empty product”
Factorials are ways to organize things.
3! = 6, because there are 6 ways to organize 3 objects:
* A B C
* A C B
* B A C
* B C A
* C A B
* C B A
How many ways are there to organize 0 objects? Well…I mean…just 1 way: an empty table. There you go; 0 objects organized. So 0! = 1.
0 is what you get when you don’t add any numbers together. 0 is the “I didn’t add anything” number. Anything plus 0 is still itself.
1 is what you get when you don’t *multiply* any numbers together. 1 is the “I didn’t multiply anything” number. Anything times 1 is still itself.
Factorials are about multiplication, so if you do the factorial of nothing, you get the “I didn’t multiply anything” number: 1.
One reason: for all numbers greater than zero, it’s true that N!/N = (N-1)!, and N!*(N+1) = (N+1)!. If 0! = 1, the pattern continues just fine, whereas picking any other number makes the behavior of the function strange.
A second reason: a factorial represents the total number of permutations possible for objects in a series. If you shuffle a deck of 52 cards, there are 52! possible resulting decks. If you shuffle a deck with 1 card, there’s only one possible order: the card by itself. If you have an empty box of cards, it likewise has a single “order”: nothing.
A third reason: even if you don’t accept the reasoning of shuffling an empty deck, there are other equations in combinatorics that only work if 0! = 1. It’s simpler to call it that by definition, than to rewrite all the combinatorics equations with special cases when one of the variables involved happens to be zero.
Wikipedia [helpfully gives a bunch of reasons](https://en.wikipedia.org/wiki/Factorial#Factorial_of_zero).
As with a lot of maths things, a bunch of things work better if we define things that way, it fits with existing conventions, and it kind of makes sense with the idea of what factorial tells us.
We *could* define it to be something else, but it wouldn’t be quite as neat.
it’s not counterintuitive at all
4! = 24 and 3! = 6 right?
to get from 4! (24) to 3! (6) we can just divide by 4.
to get from 3! (6) to 2! (2) we can divide by 3.
to get from 2! (2) to 1! (1) we can divide by 2.
and to go from 1! to 0! we can divide by 1.
1! / 1 is equal to 1
Something that I haven’t seen mentioned yet is the following. The factorial only works for positive integers. But lets say that you want a version of the factorial that works for decimals? Or negative numbers? or even complex numbers? If you want to keep all the important features of the factorial (the shape you expect and the recurrence relation [(5 + 1)! = (5 + 1)(5!) = 6(5!) = 6!] being two), then it turns out that there is only one function that works, the gamma function shifted by one, [written: Γ(n + 1)]. It turns out that Γ(n + 1) is equal to all the factorials for positive integers, and produces a value for every positive and negative real and complex number except for the negative integers, which is really cool (if you try to use the negative real integers, then you get a divide by zero). But it turns out that 0! = Γ (0 + 1) = 1. So, if you want to use the gamma function (which does some pretty cool things and is pretty simple and elegant) then you need to think that 0! = 1.
Start by thinking about a slightly more abstract problem. What happens when you combine zero things?
There are a bunch of operations that combine elements of a set. Addition can combine numbers (1+2+3=6). Multiplication can combine numbers (2*3*5=30). Concatenation can combine strings (‘ab’ • ‘c’ = ‘abc’).
Each of these operations also has a special element call the identity that doesn’t change the other thing it’s combined with. For addition it’s zero (0+x=x for all x). For multiplication it’s one. For concatenation it’s the empty string ‘’.
You can think of combining a list as starting with the identity element and the one at a time combining the elements of your list. In that sense combining a list of no elements generates the identity element. This is super intuitive for addition. Adding up the sim of no numbers is zero. Concatenating no strings together is the empty string. Similarly multiplying no numbers together is one.
As a programmer I read that title as 0 does not equal 1 and was confused why it seems counterintuitive lmao
Others have shared great answers but a diagram might also help!
[https://www.desmos.com/calculator/jjdybygnyl](https://www.desmos.com/calculator/jjdybygnyl)
This is a diagram of a generalization of factorials called the [Gamma function](https://en.wikipedia.org/wiki/Gamma_function). Anyhow, the closer you get to “0” the closer the Gamma gets to 1. And at every other integer, Gamma is the same as Factorial. It’s a smooth function that connects the Factorials.
Maybe it will confuse more than clarify but I think it’s kind of cool.
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