HomeĐời SốngWhy is 0! = 1?

Why is 0! = 1?

08:54, 28/03/2021
if you've sầu been paying particularly cthảm bại attention lớn our use of the factorial operator & & in the videos on permutations and combinations you may or may not have sầu noticed something that might be interesting so let's just Reviews factorial a little bit so if I were khổng lồ 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 lớn 1 and then I would multiply all of those things together so for example and all of this is Đánh Giá if I were to lớn say 3 factorial that's going khổng lồ be 3 times 2 times 1 if I were khổng lồ say 2 factorial that's going to lớn be 2 times 1 1 factorial by that xúc tích well that's just I just keep decrementing until I get to lớn 1 but I don't have sầu to lớn 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 lớn 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 và maybe maybe it is 0 now what we will see is that this is not the case that mathematics some mathematicians have decided và rethành viên this is what's interesting you know the factorial operation this is something that humans have sầu 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 sầu found it far more useful lớn define 0 factorial as something else lớn define 0 factorial as và there's a little bit of a drumroll here they believe sầu 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 khổng lồ permutations I'll show you why this is why this is a useful concept especially in the world of permutations và combinations which is frankly we're factorial shows up the most if I would say the most of the cases that I've sầu ever seen factorial và anything has been in situations of permutations & combinations and in a few other things but mainly permutations và combinations so let's Reviews a little bit we've sầu 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 lớn be it's going to lớn be n factorial over n minus over n minus K factorial now we've sầu also said that if we had if we had n things that we want to permute into lớn n places well this really should just be n factorial the first place has you know let's just vày this so this is the first place this is the second place this is the third place all the way you get khổng lồ the nth place well there would be n possibilities for who's in the first position or which object is in the first position và then for each of those possibilities there would be n minus 1 possibilities for which object you choose to lớn put in the second position because you've already put one inkhổng lồ 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 lớn 1 và this thing right over here is exactly what we wrote over here this is equal khổng lồ this is equal lớn 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 & then you might see why this is interesting because this is going lớn be n factorial over zero factorial so in order in order for this formula lớn apply for even in the case where K is equal lớn n even in the case where K is equal to lớn n which is this one right over here & for that khổng lồ be consistent with just plain old lô ghích zero factorial needs to lớn be equal to 1 & 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 lớn one và you keep multiplying them four zero we're just going lớn define this we're just going to define make a mathematical definition we're just going to lớn say zero factorial is equal to one and it's actually quite useful