Zero factorial or  0!

if you've been paying particularly close attention to our use of the factorial operator and and in the videos on permutations and combinations you may or may not have noticed something that might be interesting so let's just review factorial a little bit so if I were to say n n factorial that of course is going to be n times n minus not side times n minus 1 times n minus 2 and I would just keep going down until I go 2 times 1 so I would keep decrementing n until I get to 1 and then I would multiply all of those things together so for example and all of this is review if I were to say 3 factorial that's going to be 3 times 2 times 1 if I were to say 2 factorial that's going to be 2 times 1 1 factorial by that logic well that's just I just keep decrementing until I get to 1 but I don't have to decrement here I'm already at 1 so I just multiplied 1 now what about 0 factorial this is interesting 0 0 factorial so one logical thing to say well hey you know maybe 0 factorial is 0 it's you know I'm just starting with itself it's already below 1 and maybe maybe it is 0 now what we will see is that this is not the case that mathematics some mathematicians have decided and remember this is what's interesting you know the factorial operation this is something that humans have have invented that they think is just an interesting thing it's a useful notation so they can define what it does and math mathematicians have found it far more useful to define 0 factorial as something else to define 0 factorial as and there's a little bit of a drumroll here they believe 0 factorial should be 1 and I know based on the the the reasoning the conceptual reasoning of this this doesn't make any sense but since we've already been exposed a little bit to permutations I'll show you why this is why this is a useful concept especially in the world of permutations and combinations which is frankly we're factorial shows up the most if I would say the most of the cases that I've ever seen factorial and anything has been in situations of permutations and combinations and in a few other things but mainly permutations and combinations so let's review a little bit we've said that hey you know if we if we have n things and we want to figure out the number of ways to permute them into K spaces it's going to be it's going to be n factorial over n minus over n minus K factorial now we've also said that if we had if we had n things that we want to permute into n places well this really should just be n factorial the first place has you know let's just do this so this is the first place this is the second place this is the third place all the way you get to the nth place well there would be n possibilities for who's in the first position or which object is in the first position and then for each of those possibilities there would be n minus 1 possibilities for which object you choose to put in the second position because you've already put one into that position now for each of these n times n minus 1 possibilities where you've placed two things there would be n minus 2 possibilities of what goes in the third position and then you would just go all the way down to 1 and this thing right over here is exactly what we wrote over here this is equal to this is equal to n factorial but if we directly applied this formula if we applied this formula this would need to be n factorial over n minus n factorial and then you might see why this is interesting because this is going to be n factorial over zero factorial so in order in order for this formula to apply for even in the case where K is equal to n even in the case where K is equal to n which is this one right over here and for that to be consistent with just plain old logic zero factorial needs to be equal to 1 and so the mathematics community has decided hey this thing we've constructed called factorial you know we said hey you put an exclamation mark behind something in all of our heads we say you kind of countdown that number all the way to one and you keep multiplying them four zero we're just going to define this we're just going to define make a mathematical definition we're just going to say zero factorial is equal to one and it's actually quite useful