Not really. A great teacher would have put the formula into context and made it more accessible. And don't go saying that it should've been understood from the start because then what's the teacher for?
For people who dont know why factorials calculate arrangements, this is how my teacher explained it that i thought was really good: So imagine we have 3 counters. Red, blue, and green. We need to arrange them, and we do so by selecting one at a time. For the first selection, there are 3 possibilities, one for each colour. On the second round, there are 3 possibilities, minus the one we already chose. So 3-1=2 possibilities. If you remember, we find the total number of outcomes by multiplying the number of outcomes from each stage together, say when you toss a coin twice there are two outcomes for each stage, so 2*2 outcomes, which is 4. HH, HT, TH, TT. We do that here. So when we do our final stage, there is only one choice, so our total outcomes is 3*2*1, or 3!
Finished calculus 3 and just found out factorials are how many ways you can arrange that many things. I don't know how I never mentally connected those
@@davidwu8951I learned about factorials in the context of probability calculation and I still only now figured that out thanks to the video. I finished school in 2018
This is the style of teaching that's straight to the point that would've made me actually put effort in my calculus classes. This makes it accessible, fun, and memorable. In 3 minutes I properly learned about factorials and subfactorials, and can sum them up for a random person on the street. And the best part is I'm confident that I'll remember the concept years from now just because of this explanation!
This is the sort of thing I'm delighted to learn exists, especially that there's a closed form. Also, your calculated example was super-pedantic, which I really appreciate, because if I tried the closed form on my own, I'd probably make an arithmetic error :( Thanks!
Thanks for making these videos! This was so easily understandable, I used to sit in Probability class and finish the session without understanding a single thing SMH, really wish I had access to youtube back then, would've done so much better in math and physics subjects.
Never heard them being called 'sub factorials' before. In my 11th grade maths class, we call this 'Disarrangement', but its the same thing. Cool to know that it is called this too! Will definitely info-drop this with my classmates!
A very frequently asked question based on sub factorials (derangement) that is asked in many aptitude exams in India is this - Suppose there are 5 letters and 5 envelopes. In how many ways can you put the letters in the envelopes so that none of the letters reach its intended destination. The answer to this problem is simply !5, which is 44. Great video Andy ;)
Thank you! I learned something new. I've approached problems that were described by this in my work but never knew how to describe it. I'd just solve it the long way in Excel.
I see many people who know calculus being surprised by the use of factorials in arrangement of stuff. I'm curious, were you all not taught permutations and combinations simultaneously, before or after calculus?
I covered factorials when I learned about series in calculus. However, I didn’t cover permutations and combinations until I got to discrete math in college.
@@K1JUY Interesting, though I can see how teaching only upto Taylor series would be sufficient for basic calculus, though for me P&C was taught before calculus so that our algebraic grasp would be concrete.
Great explanation! The very first time I ever heard of factorals was in an explanation that if you shuffle a deck of cards you are very likely to have been the very first person to have shuffled that combination. As I recall it was !51, which is an unimaginably large number. Had these fun factoids or an explanation as succinct as yours been in my high school I might have been more interested in the subject.
I've never knew about Subfactorials, that's really cool. I'm curious on use cases for it? When would I want to eliminate an arrangement that has items in already matched positions? Obviously, math is based on the abstract generic usage, but I'd love to see an example (word problem) of Subfactorial.
As someone with only a high school understanding of math, the subfactorial topic is neat and all, but seeing someone finally explain what ∑ means is probably invaluable. Thank you. It means 'add everything between the number under ∑ and the number over ∑,' right? Did I interpret that correctly?
Yes, you are correct. You start from whatever the variable under the sigma, in this case k, is equal to and you substitute that value of k into the equation that comes after the sigma. when you have this, you increment k by 1 and do the same thing to get a new number and add the two numbers together. Keep on doing this until your value of k matches the number above the sigma.
Damn, calculus is amazing. A shame I never learned it at schol because somehow, my country decided it's not important to be teached at high school. This shit is awesome
Your spirit is really amazing but unfortunately this isn't calculus😅 if you want there are tons of resources online for free to study calculus and multivariable calculus you can actually get Full courses (with exams and assignments and lectures and sections...etc) from MIT Open courseware
It's not calculus, it's combinatorics. Also there's a lot of people who won't use calculus concepts directly in their lives, so it would be pointless to teach it at high schools. For us that do like math, we can always use the internet to learn more stuff than what is taught in the school.
Back when I went to school, this was covered in Discrete Mathematics. I know we also covered it in high school, but it might have just been a general advanced math class? Combinatorics can serve a purpose in common life situations (ok, not super common, but still useful at times).
@@nech060404 Everybody uses calculus in the sense that it is necessary to engineer the devices we use in our daily lives. Not everybody have to know how to calculate an integral though, just like not all mathematicians have to know what was the Nanjing massacre, how to speak portuguese or how to improve a website SEO. Different jobs for different people requires different skills.
i put this on my watch later list when I got this video recommended to me (which was not long after it released) but never watched it. Now that I did, I don't know why I didn't do it earlier. Pretty neat
I had not heard of subfractionals and went in deep after watching this video. And of course, a wild e appeared. The limit as n approaches infinity of !n/n! is 1/e. How exciting.
Brit in the UK. Despite having A-Level maths and doing the first year of an Astrophysics degree, before switching to Chemistry. This is the first time I've heard of subfactorials. Thank you for the fascinating video.
Yeah im finishing my physics degree this year and I have genuinely never heard of them. Perhaps they're not important to my specialization, or simply i've been using derivations. real cool thing to know, though.
Can you please use a black (or dark grey) backhround and white (or light grey) text? It would be much easier to look at the screen. Thank you, and keep up the good work )
If n-->∞ (tends to infinity) then lim n-->∞ (!n = n!/e) Meaning that, If the 'n' is large enough then the sub factorial of 'n' OR '!n' is approximately equal to 'n!/e' Is this right?
I have BA in mathematcs and I just learned something. I also enjoyed your clear presentation - subscribing! (No, that's not the factorial of "subscribing")
I did it like this: n! is Γ(n+1) = Γ(n+1, 0) for n being a natural number. (I always say it's equal, but the definition says it's not. qwq) !n is Γ(n+1, -1)/e. Γ(n, x) is the incomplete gamma function which is defined as the integral from x to infinity of t^(n-1)*e^-t dt. For odd n and negative t, t^(n-1)*e^-t is negative. when n=3 and t
But isn't it kind of weird, how the Factorial counts the original ABC-permutation, whereas the subfactorial doesn't? So, at least from the verbalexplanation, I feel like !3 should be 3, not 2
How many ways can you arrange the individual letters A, B, and C? 6 ways, one of the ways is ABC. How many ways can you scramble the string of letters "ABC"? Only 2 ways because "ABC" is not a scrambled version of "ABC."
@@ManifestedMadness Neat! But since that I think my formula was wrong, I guess. Like I used 3 ÷ (3 - 1) ÷ (3 - 2) and got 1.5, but when I used your formula (1/3!) and got 1.666666667.
If you want to compute it quickly, just round n!/e to the nearest integer. (Which tells you also that a random permutation has about 1/e chances to have no fixed point.)
Subfactorials count the derangements of a list of items. Derangements are the permutations of the items when each item is out of its original place. Lets say you have a list ABCD. So a derangement of those items will count the permutations when A is not on first place and B is not on second place and C is not on third place and D is not on fourth place. The derangements of ABCD are BADC BCDA CADB CDAB CDBA DABC DCAB DCBA
I was just messing around with this using an online calculator, and it seems like if you do x!/!x it will always get closer and closer to Eulers number. Like 7!/!7 = 2.71… not exactly eulers number but close. Same for 10!/!10. You may have stated this in this video although I thought it was cool!
![Why you didn't learn tetration in school[Tetration]](/img/n.gif)
Never heard of sub factorials before, very fun!
exciting*
@@LiriosyMas you're right, I can't believe I made such a rookie mistake!
Me too
yeah!
how exciting
Subfactorials basically tell you how many different ways you can completely re-arrange a set of objects
thanks, that explanation is way more clear
What do you mean by ‘completely’?
@@alex.g7317 such that no object remains in its original position.
@@The_Story_Of_Us ah, right… I always wondered what use having sub factorials can have. Do you know any uses?
@@alex.g7317 I’d only be guessing the obvious really.
How exciting
How exciting
How exciting !
How exciting!
This comment looks important so let’s put a box around it
How exciting
hard time learning math? this guy helps u by explaining almost every equation and formula and gives examples of it. overall 5 stars math teacher
:)
Not really. A great teacher would have put the formula into context and made it more accessible. And don't go saying that it should've been understood from the start because then what's the teacher for?
Calm down br @@denhurensohn9276
For people who dont know why factorials calculate arrangements, this is how my teacher explained it that i thought was really good:
So imagine we have 3 counters. Red, blue, and green. We need to arrange them, and we do so by selecting one at a time. For the first selection, there are 3 possibilities, one for each colour. On the second round, there are 3 possibilities, minus the one we already chose. So 3-1=2 possibilities. If you remember, we find the total number of outcomes by multiplying the number of outcomes from each stage together, say when you toss a coin twice there are two outcomes for each stage, so 2*2 outcomes, which is 4. HH, HT, TH, TT. We do that here. So when we do our final stage, there is only one choice, so our total outcomes is 3*2*1, or 3!
Well explanation but I didn't understand a sht may be my English weak
@@Allena_boofe is it your second language? feel free to ask me any questions abt it im happy to try explain differently.
@@meks039 yes please explain me if you can
@@meks039 it would be very greatful for me
@@Allena_boofe okay so is there anything specific you dont quite get? just copy paste in the bits where you lost track if you dont get it.
Finished calculus 3 and just found out factorials are how many ways you can arrange that many things. I don't know how I never mentally connected those
Not sure if you’ve ever used factorials for calculating probability but it’s a way to closely connect the two!
@@davidwu8951I learned about factorials in the context of probability calculation and I still only now figured that out thanks to the video. I finished school in 2018
It was in discrete math (or combinatorics - seen it called both in different schools) where I learned that
I literally used them for a chapter in combinatrics wnd never realised.
So THAT’S why 0! is equal to 1. Mind blown
This is the style of teaching that's straight to the point that would've made me actually put effort in my calculus classes. This makes it accessible, fun, and memorable. In 3 minutes I properly learned about factorials and subfactorials, and can sum them up for a random person on the street. And the best part is I'm confident that I'll remember the concept years from now just because of this explanation!
I’ve never thought about factorials as arranging things. Cool way to think of it. Thanks for the informative vid man
Your simple style, fun equations, and obvious interest in math made me subscribe 💯
You mean exciting
There is no fun in math, only an abyss
This is the sort of thing I'm delighted to learn exists, especially that there's a closed form.
Also, your calculated example was super-pedantic, which I really appreciate, because if I tried the closed form on my own, I'd probably make an arithmetic error :(
Thanks!
Thanks for making these videos! This was so easily understandable, I used to sit in Probability class and finish the session without understanding a single thing SMH, really wish I had access to youtube back then, would've done so much better in math and physics subjects.
I don’t know if I ever learned this, but very fascinating. Thanks for the knowledge
Your explanation are very exciting! Thanks to you, I finally understand Summations!!! Thank you!!!
Dang! Clear and clean explanation. No fluff, no carryon. Nice. 👏
Math can be really fun if explained properly. I wish I had a teacher like you when I was learning things.
Never heard them being called 'sub factorials' before. In my 11th grade maths class, we call this 'Disarrangement', but its the same thing. Cool to know that it is called this too! Will definitely info-drop this with my classmates!
You gotta admit that 'derangement' sounds funnier.
Well we call it 'dearrangement' dk if it's a word or not tho
you are by far the best teacher
A very frequently asked question based on sub factorials (derangement) that is asked in many aptitude exams in India is this -
Suppose there are 5 letters and 5 envelopes. In how many ways can you put the letters in the envelopes so that none of the letters reach its intended destination.
The answer to this problem is simply !5, which is 44.
Great video Andy ;)
Its really appreciable someone teaching maths in terms of how its used.
Are there any applications for subfactorial?
Thank you! I learned something new. I've approached problems that were described by this in my work but never knew how to describe it. I'd just solve it the long way in Excel.
I see many people who know calculus being surprised by the use of factorials in arrangement of stuff. I'm curious, were you all not taught permutations and combinations simultaneously, before or after calculus?
I covered factorials when I learned about series in calculus. However, I didn’t cover permutations and combinations until I got to discrete math in college.
♥️♥️
With love
@@K1JUY Interesting, though I can see how teaching only upto Taylor series would be sufficient for basic calculus, though for me P&C was taught before calculus so that our algebraic grasp would be concrete.
Why does the subfactorial formula's sum start from 0 instead of 2?
In the formula you can also start at k=2 for any !x where x>1 just because the first two terms always cancel out.
Great explanation! The very first time I ever heard of factorals was in an explanation that if you shuffle a deck of cards you are very likely to have been the very first person to have shuffled that combination. As I recall it was !51, which is an unimaginably large number. Had these fun factoids or an explanation as succinct as yours been in my high school I might have been more interested in the subject.
first time hearign about subfactorial but this was pretty cool and kept my attention throughout
I've never knew about Subfactorials, that's really cool. I'm curious on use cases for it? When would I want to eliminate an arrangement that has items in already matched positions? Obviously, math is based on the abstract generic usage, but I'd love to see an example (word problem) of Subfactorial.
This is new to me and very interesting.Thanks Andy
I knew about combinations and permutations but not this secret third thing. Neat!
Freaking cool, bro! I’m gonna use these things in Scholars Bowl 😂
Never knew i was a math nerd until i started seeing ur videos on insta and now im here. How exciting
Easily explained a bit of permutations and derangements too!
Great😊
Really commendable 🎉
As someone with only a high school understanding of math, the subfactorial topic is neat and all, but seeing someone finally explain what ∑ means is probably invaluable. Thank you.
It means 'add everything between the number under ∑ and the number over ∑,' right? Did I interpret that correctly?
Yes, you are correct. You start from whatever the variable under the sigma, in this case k, is equal to and you substitute that value of k into the equation that comes after the sigma. when you have this, you increment k by 1 and do the same thing to get a new number and add the two numbers together. Keep on doing this until your value of k matches the number above the sigma.
yep. Its a sum :)
Man this was amazing!!, loved the video
Factorials are very useful in a number of situations, like probability, sorting, etc. What is the use of subfactorials?
I am interested too.
The way you teach me is really awesome man ❤
very nice ! today i have learned sth new. thanks sir
the factorial explanation made me drop the like best way to explain what's a factorial
The fact that he is so cute and pretty makes his videos so much better
So 8 years of Andys Math videos. How exciting.
Would've been hilarious if the video ended at 0:07 lmao
Best explanation I've seen for this - Good job Mr. Math.
Damn, calculus is amazing. A shame I never learned it at schol because somehow, my country decided it's not important to be teached at high school. This shit is awesome
Your spirit is really amazing but unfortunately this isn't calculus😅 if you want there are tons of resources online for free to study calculus and multivariable calculus you can actually get Full courses (with exams and assignments and lectures and sections...etc) from MIT Open courseware
It's not calculus, it's combinatorics. Also there's a lot of people who won't use calculus concepts directly in their lives, so it would be pointless to teach it at high schools. For us that do like math, we can always use the internet to learn more stuff than what is taught in the school.
Back when I went to school, this was covered in Discrete Mathematics. I know we also covered it in high school, but it might have just been a general advanced math class? Combinatorics can serve a purpose in common life situations (ok, not super common, but still useful at times).
@@Israel220500 I disagree we should require everyone to use calculus. Calculus is the study on how things change in systematic ways.
@@nech060404 Everybody uses calculus in the sense that it is necessary to engineer the devices we use in our daily lives. Not everybody have to know how to calculate an integral though, just like not all mathematicians have to know what was the Nanjing massacre, how to speak portuguese or how to improve a website SEO. Different jobs for different people requires different skills.
Missed opportunity to talk about other proofs for 0! = 1, but i guess they might end up in another video. That would be very
exciting
What a pitty
How exciting
Eddie Woo has made a video about that
I thought that it's gonna be a bigger version of factorials like [ exponentiation --> tetration ], but ok I learned something.
i put this on my watch later list when I got this video recommended to me (which was not long after it released) but never watched it.
Now that I did, I don't know why I didn't do it earlier.
Pretty neat
The 1st time I learned factorials was in ICS 111 @ Honolulu Community College decades ago.
that explanation was really easy to follow!
I had not heard of subfractionals and went in deep after watching this video. And of course, a wild e appeared. The limit as n approaches infinity of !n/n! is 1/e. How exciting.
figures.
i like math but i’m not good at it, so desmos’ graphing calculator is a good friend of mine.
so i randomly did !x/x! and silently cried
Dude I just did the same thing but the other way around. That actually kinda funny lol.
I learned so much in this video, you have no idea.
Finely understanding why factorial 0 == 1, because of arrangements of course !!! Good explanation man, thank's a lot. 👍👍👍.
Why is the original configuration not counted towards the subfactorial?
This is something they never mentioned to me at school. Fascinating! 👍
Why was ABC arrangement not part of the subfactorial of 3.
What a great recursive formula for derangement. reminds me of dynamic programming techniques.
Ahhh yes discreet mathematics
Interesting. But, what's the practical use of subfactorials?
Thanks! First mathematical explanation on sub-factorial
Your are a damn good teacher😂 thanks man
You explained so clearly. Thank you. It was interesting!
FOR ANYONE WHO CARES N FACTORIAL DIVIDED BY N SUBFACTORIAL IS EQUAL TO E
Amazing! I have never ever heard of this before.
I love all math and I’ve never heard of a subfactorial. Makes perfect sense . Thx
My mind is blowing, this is so exciting!
Brit in the UK. Despite having A-Level maths and doing the first year of an Astrophysics degree, before switching to Chemistry. This is the first time I've heard of subfactorials. Thank you for the fascinating video.
Yeah im finishing my physics degree this year and I have genuinely never heard of them. Perhaps they're not important to my specialization, or simply i've been using derivations. real cool thing to know, though.
Mathematicians: Uhh its too long to write.. let's shorten it!
*Random TH-camr: Content!!!*
Exciting....so much exciting!
Can we write 5!5=?
You probably need to use parentheses
Never heard of that before, where is it used?
I did all the advanced level maths in high school. During finite math (combinations and permutations) we were never told about subfactorials.
Is it the formula that is used to count dearrangement in enclosing n letters in n envelopes
Can you please use a black (or dark grey) backhround and white (or light grey) text? It would be much easier to look at the screen. Thank you, and keep up the good work )
If n-->∞ (tends to infinity) then
lim n-->∞ (!n = n!/e)
Meaning that,
If the 'n' is large enough then the sub factorial of 'n' OR '!n' is approximately equal to 'n!/e'
Is this right?
What about the Spanish factorial, for example, 5¡ ? Note the upside down factorial sign.
What use do subfactorials have?
What is the website or whatever that you’re using for this?
Cool video but just wanted to say because I realised it and can't unsee it, your outfit looks almost exactly like Terry Davis
Subfactorials are so cool. Can you explain Tetration too?
I have BA in mathematcs and I just learned something. I also enjoyed your clear presentation - subscribing! (No, that's not the factorial of "subscribing")
Awesome, thank you!
Thank you, now i know the principles of sum too 😂😂😂
So we are taking the initial arrangement of these objects and we make cycles?
This is actually a good piece of knowledge to have, might be useful one day
I did it like this:
n! is Γ(n+1) = Γ(n+1, 0) for n being a natural number. (I always say it's equal, but the definition says it's not. qwq)
!n is Γ(n+1, -1)/e.
Γ(n, x) is the incomplete gamma function which is defined as the integral from x to infinity of t^(n-1)*e^-t dt.
For odd n and negative t, t^(n-1)*e^-t is negative. when n=3 and t
But isn't it kind of weird, how the Factorial counts the original ABC-permutation, whereas the subfactorial doesn't? So, at least from the verbalexplanation, I feel like !3 should be 3, not 2
I was also thinking the same thing.
Someone please answer this question
How many ways can you arrange the individual letters A, B, and C? 6 ways, one of the ways is ABC.
How many ways can you scramble the string of letters "ABC"? Only 2 ways because "ABC" is not a scrambled version of "ABC."
Reminds me of a free group action. All the nonidentity permutations are derangements.
This reminds me of the method to finding the determinant of a matrix, where the terms change positivity, are they related in sime combinatorial way?
A simplier way to calculate it without Sigma: *[k! - (k)^2 + 1] × (-1)^k*
Subfactorial is a number of cyclic shift, isn't it?
Or with something more than 3 it won't be the case?
Is there a anti-factorial, like 3¡? Where it's like 3 ÷ 2 ÷ 1? With the formula being n ÷ (n - 1) ÷ (n - 2)... = a?
You could just do 1/x! Or if you want to divide the first term by all other terms x^/x!
@@ManifestedMadness Neat! But since that I think my formula was wrong, I guess.
Like I used 3 ÷ (3 - 1) ÷ (3 - 2) and got 1.5, but when I used your formula (1/3!) and got 1.666666667.
Is there any proof for subfactorial formula,or this formula came out of intuition.
How exciting - indeed? And yet, you made it interesting.
If you want to compute it quickly, just round n!/e to the nearest integer. (Which tells you also that a random permutation has about 1/e chances to have no fixed point.)
Exciting
I wonder if there any other videos where there's a figure in the doorway?
I haven’t needed to know this since 2002 or something. Why is this so interesting? I won’t need it again until my kid asks me math questions.
Excellent!!
I can't even start to imagine Grahams number factorial.
Subfactorials count the derangements of a list of items.
Derangements are the permutations of the items when each item is out of its original place.
Lets say you have a list ABCD.
So a derangement of those items will count the permutations when A is not on first place and B is not on second place and C is not on third place and D is not on fourth place.
The derangements of ABCD are
BADC
BCDA
CADB
CDAB
CDBA
DABC
DCAB
DCBA
I made this in scratch over summer. Pretty fun project.
I was just messing around with this using an online calculator, and it seems like if you do x!/!x it will always get closer and closer to Eulers number. Like 7!/!7 = 2.71… not exactly eulers number but close. Same for 10!/!10. You may have stated this in this video although I thought it was cool!