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@kilgorezer24 күн бұрын
i agree, that was fun
@disnecessaurorex490824 күн бұрын
I feel like "It's 1/e, isn't it?" is the "he is right behind me, isn't he?" of maths
@RickofUniverseC-13724 күн бұрын
Great analogy!
@CliffSedge-nu5fv23 күн бұрын
Euler is always there. Sometimes he's downstairs.
@blasphemer_amon15 күн бұрын
Was not expecting to see a 2021 Abu Dhabi reference here 💀
@Asiago913 күн бұрын
It's always an expression to do with pi, e, or ln of something it feels like in these higher level math videos
@apokalypthoapokalypsys957311 күн бұрын
@@blasphemer_amonit's not a reference to "2021 Abu Dhabi", it is a reference to the thousands of badly written movie scripts with tiresome clichés
@enderslice837824 күн бұрын
IT'S ALWAYS e LEAVE ME ALONE EULER
@patrickwienhoft798724 күн бұрын
I love that even in your furious use of caps lock you still have the respect for mathematics to properly write e as lower case
@enderslice837824 күн бұрын
My rage doesn't make me disregard how symbols work. I have standards.
@pagjimaagjinen973324 күн бұрын
This smaller case e makes it seem like you say e calmly, and shout the rest
@kingnoob350324 күн бұрын
Euler will chase you for your life
@interconnected.24 күн бұрын
I love this reply lmao 🤣
@deepdrag813124 күн бұрын
“A dark room with several men who aren’t wearing any hats.” Oh! The nightmares I’ve had about that!!
@WrathofMath24 күн бұрын
😂😂
@McDonalds-Empty-Cup24 күн бұрын
Nightmares? I had fantasizes about that
@-danR24 күн бұрын
That's Inception level 1 The deeper nightmare is none of them getting their hats back. This is a branch of Applied Mathematics I never knew I needed.
@CliffSedge-nu5fv23 күн бұрын
Oh, _hats_ not wearing hats, right. That's what they aren't wearing.
@davidwright843223 күн бұрын
It might not be the lack of hats in such a situation that's hazardous, but the lack of clothing well south of the hat region. Or if not hazardous, perhaps adventurous. e ...xactly!
@Frownlandia24 күн бұрын
I have a feeling the men without hats are happy to be partying in a dark room. After all, they can dance if they want to, and they can leave their friends behind.
@WrathofMath24 күн бұрын
True, but one has to worry about the hats going wherever they want to, a place the men may never find
@davegrimes338522 күн бұрын
And, if they don't dance, well they're no friends of mine
@madacsg15 күн бұрын
It will be definitely a SAFETY DANCE! 😉🤗😄 kzbin.info/www/bejne/Z6HCc6mXfdiafck
@spectria.limina13 күн бұрын
I'm glad these men are so safety-conscious while dancing.
@carly09et24 күн бұрын
This is Weird : I was just doing some economic modelling and this result popped out. This explains the entropy of a market, as it estimates states.
@WrathofMath24 күн бұрын
That sounds very interesting, will have to research
@joshuaiosevich372724 күн бұрын
You might find it useful that derrangements can be represented by an integral: int_{0}^{/infty} (e^(-x))(x-1)^ndx
@carly09et24 күн бұрын
@@joshuaiosevich3727 That's true. But I was finding the ratio by construction - ie a fractal result. This explained why the result converged on this.
@815TypeSirius10 күн бұрын
@@joshuaiosevich3727 linear algeba ruins everything
@joshuaiosevich372710 күн бұрын
@@815TypeSirius I'm afraid I'm too dense to get what you're saying.
@Ryanisthere24 күн бұрын
0:38 this is getting very spicy several men without hats
@WrathofMath24 күн бұрын
Spicy indeed, in a world of hatless men, where do we find God?
@the_siili24 күн бұрын
@@WrathofMaththe God is a man without a hat
@tweer6424 күн бұрын
Did they leave their friends behind?
@Akio-fy7ep23 күн бұрын
@@tweer64 If their friends don't dance, well they're ... no friends of mine.
@der.Schtefan24 күн бұрын
A dark room with many men in it... Sounds like a Berlin nightclub.
@empathogen7524 күн бұрын
And then e showed up.
@nicholasiverson978423 күн бұрын
@@empathogen75 only if it was one of Those clubs, with hatless men.
@coyets23 күн бұрын
It does indeed sound like a Berlin nightclub, but the calculation was not gender specific, and it could therefore also be applied to any dark room where people remove their hats on entering.
@matthiasschulz356917 күн бұрын
I feel like there's a Flight of the Conchords song about this kind of situation ...
@8bits5911 күн бұрын
@@coyetsyes, obviously. Did you miss the joke?
@ReinOfCats24 күн бұрын
I'm a game developer, and a strikingly similar scenario - and result - came up awhile back when doing a deep dive on some item drop rate adjustments. Imagine you have a monster that drops an item when defeated at a rate of 1 in 100 times and then you defeat 100 of that monster. What's the chance you've gotten at least 1 of that item? Due to the "at least 1", this is easier to count the inverted result of "how many times did you fail to get the item" and repeat 100 times. So: (1-1/100)^100. And then invert that result: 1 - (1-1/100)^100. Giving a result of approximately ~63.4% chance of getting at least once. Generalizing this as n instead of 100, and then letting an n approach infinity, we get the result: Lim n->inf [1 - (1 - 1/n)^n] = 1 - (1 / e)
@Dexaan23 күн бұрын
I remember mentally calculating critical chance for multiple critical chances way back in the Warcraft 3 days and my rough math always seemed to end up near either 33% or 66%
@Alphabetatralala2 күн бұрын
Geometric distribution is a bitch.
@whamer10024 күн бұрын
as a programmer, seeing "!n" just makes me think "logical not n" which evaluates as either 0 or 1 depending if its non-zero lmao
@cosmnik47224 күн бұрын
bitwise not also works
@dapcuber722524 күн бұрын
I'd be more inclined to think bitwise not as I usually write logical not as ¬ but yeah this would be confusing 😭
@kuwi106124 күн бұрын
@@cosmnik472 For that ~ is used like this ~n
@litfill5424 күн бұрын
or strict n in haskell
@JdeBP24 күн бұрын
Those who have programmed in some home computer BASICs will be thinking "pling n".
@KookyPiranha24 күн бұрын
this is literally how i process every combinatorics problem hoping all the terms cancel out when they dont
@meatyman480324 күн бұрын
Finding this guy in a math video is a fever dream
@James221024 күн бұрын
Brings a whole new meaning to "Statements dreamed up by the utterly Deranged" (from the "stop doing math" meme)
@furrball24 күн бұрын
I wasn't wearing socks and my toes blew up.
@WrathofMath24 күн бұрын
Sorry 😞
@VioletRM24 күн бұрын
a small price for science
@furrball23 күн бұрын
@@WrathofMath np, that solved having to trim toenails.
@Thk1018896522 күн бұрын
Unfortunate
@Dexuz18 күн бұрын
Your fault for having toes
@briancooke425924 күн бұрын
Your pace and depth are perfect. I would not attempt those formulas on my own, but you made perfect sense if them. Thank you so very much!
@Kapomafioso22 күн бұрын
11:00 just a note: I think here either the sum should be marked with i < j (not i =/= j), or, if written in this way, there should be 1/2 in front of it. You don't want to repeat elements: A1 intersection A2 and then A2 intersection A1. But the result is correct, the non-repeating sum over i < j is equal to n choose 2.
@scottmiller259115 күн бұрын
Came here to say this. The n choose 2 fixes this, since it ignores permutations of the choosing, which is why even though the left hand side is wrong, the right hand side is correct. This also means the higher order indices in the summation later on for the inclusion exclusion formula need to be written as i < j < k ... for all free indices from the set, or put another way 1 =< i < j < k ... =< n.
@Frahamen24 күн бұрын
that's quite a deranged equation.
@WrathofMath24 күн бұрын
You could say that
@asmithgames592624 күн бұрын
1/e also shows up in another famous math problem, which I'll poorly paraphrase: When dating, what % of the total pool should you check out before committing to one? The answer is 37% of the pool, 1/e.
@WrathofMath24 күн бұрын
With how big dating pools are now due to modern transportation and dating apps, I still have a significant amount of work to do 🤣
@robo300720 күн бұрын
It also marks the point where the graph y = x^x stops decreasing and starts to increase
@incription24 күн бұрын
In my head I thought "its definitely something like e, or 1/e", and imagine my surprise when I saw the result! Although, not much of a surprise, whenever probability is involved, e will show up.
@WrathofMath24 күн бұрын
Yeah, for the mathematically initiated, it may be far less surprising. but it’s so amusing to take a silly word problem like that regarding hats, and the answer is 1/e 😂
@yan-amar24 күн бұрын
What I love about mathematicians is that they'll mention Greek gods like it's 1200 BC.
@thedoublek481622 күн бұрын
When the Wrath of Zeus meets the Wrath of Math. "Our fight will be legendary!"
@Hyreia10 күн бұрын
"A derangement" is such a hilarious term for something in math. I love it.
@geoffstrickler12 күн бұрын
Great explanation, you show how it’s calculated, how it’s relevant, and the end result is actually simple to calculate.
@WrathofMath12 күн бұрын
Thank you!
@Connorses24 күн бұрын
i was scared for just a second when you started drawing that hat rack
@-danR24 күн бұрын
I was wondering how he would manage to hang the hats on it.
@KylerRaineP.Nangit7424 күн бұрын
6:39 WHAT YOU'RE LETTING AI?!?!?!
@JuniperHatesTwitterlikeHandles24 күн бұрын
AI's taking everyone's jobs, not even the set of all permutations where i is fixed is safe....
@WrathofMath24 күн бұрын
gotta do what you gotta do
@blacklistnr124 күн бұрын
I appreciate the gentle transition to combinatorics via hats and Zeus's wrath :))
@dukenukem977024 күн бұрын
I’m excited to integrate a “deranged“ math lesson into my sons home-study curriculum!
@lollol-tt3fx24 күн бұрын
poor son😂
@isavenewspapers889024 күн бұрын
Oh cool. I love taking an integral of the subfactorial function.
@TamWam_3 күн бұрын
i'm excited to show this to my math teacher
@IRLtwigstan24 күн бұрын
I remember finishing our subfactorials a year ago and I loved them so much. I made a scratch project to plug a number in to give the sub factorial of the number.
@WrathofMath24 күн бұрын
Super fun!
@IRLtwigstan23 күн бұрын
@ Hell yeah!
@AusTxMale24 күн бұрын
That was well worth the wait. Thanks for such a fun explanation.
@NestorCustodio24 күн бұрын
I will never write a "3" or a "2" as legibly as this man did @ 2:30. 😢
@WrathofMath24 күн бұрын
I was cooking with those
@CliffSedge-nu5fv23 күн бұрын
I did once, and then never again. 😢
@terrybull379820 күн бұрын
However; not equal looked indistinguishable from +/- @11:30 :)
@stephenspackman557324 күн бұрын
Fixed points in this sense are cycles of length 1. The obvious generalisation is to permutations with other minimum (and indeed maximum) cycle lengths. These would be practical things to know.
@Dagobah35914 күн бұрын
5:34 Missed opportunity to say it would blow our hats off. Booo!
@WrathofMath14 күн бұрын
I will not be tempted by your cheap puns!
@MPBSODIYL23 күн бұрын
As a fascinating exercise, consider the following: Suppose you help n people receive their own hats by randomly distributing all of them. Then, those with incorrect hats return them to you to randomly distribute again among those still missing hats. What is the expected number of iterations it will take for everyone to have their own hat? (I more or less have a proof for this, but i also have homework due today 😢)
@Dexaan23 күн бұрын
I bet the proof wouldn't fit in the margin, either.
@danredebeer9 күн бұрын
is the expectation not just 1/n! ?
@NexusSpacey15 күн бұрын
This music is making me think I'm watching a Zullie the Witch video
@jonathanallan500724 күн бұрын
I think that if we asked 10000 people the question (for some arbitrary number of hats, like 20) to give a percentage from 0-100 that no-one gets their hat back there would be quite a peak at 37%. Yeah, there would likely be one at 73% too.
@bitonic58922 күн бұрын
Omfg it's really everywhere
@Kapomafioso22 күн бұрын
Why would there be a peak at 73% as well?
@jonathanallan500721 күн бұрын
@@Kapomafioso Second most popular "random" number from 0 to 100.
@ruin13074 күн бұрын
Love how this is done sooo much easier with like a for loop and if statement in programming
@gerald5619 күн бұрын
Thanks for the presentation! This is a clssical gem of discrete mathematics.
@mab93168 күн бұрын
Beautiful explanation. Thank you.
@George-tk2hjКүн бұрын
This was really fun, and the notation wasn’t too difficult. Thanks!
@truebino23 күн бұрын
The night theme of Hateno village made this an emotional hat story for me...
@WrathofMath23 күн бұрын
Takes me back to sophomore year of college
@davidbelk4622 күн бұрын
So, if you need to calculate a subfactorial for some reason and you wish to save a lot of time, just divide the factorial by the number e then round to the nearest integer. It works every time.
@zacjdavis24 күн бұрын
10:21 I hear faint sounds of skyward sword. Haha
@luinerion24 күн бұрын
5:38 "it will blow your socks off" What a relief that it won't blow my _hat_ off 😌
@WrathofMath24 күн бұрын
The hats are long gone at this point 😂
@NoriMori199214 күн бұрын
Enjoyed the SM64 music 😊
@SquidLikesTalking24 күн бұрын
What a fantastic video, such an enjoyable watch, the mario 64 music was just the cherry on top
@WrathofMath24 күн бұрын
Thanks so much! Trying to get the music at the right volume, I think I got it just about right this time.
@MVRX.6 күн бұрын
Great explanation!
@WrathofMath6 күн бұрын
Thank you!
@kylesnotepic2 күн бұрын
that's called an admin command
@francocatanzaro9618 күн бұрын
4:23 I hear your Easter Egg of putting the Select File Theme from SM64, as Pannenkoek does! Very clever!
@denischen819624 күн бұрын
Is there a math symbol or function for how many permutations of n have m fixed points where 0
@cocoabutter588824 күн бұрын
Something like n-m choose n maybe?
@tomkerruish298225 күн бұрын
Clearly, they were not doing the Safety Dance. (Gen X earworm, activate!)
@surkh24 күн бұрын
Well, then they're no friends of mine!
@lulairenoroub386924 күн бұрын
But they were men without hats And they were dancing And they were friends The men without hats never claimed they didn't "own" hats. They just didn't have any, currently, as is the case with our dancers
@porl4224 күн бұрын
I had exactly the same thing in my head 😂
@RandyKing31424 күн бұрын
i was trying to work out a comment like this…well played!
@topquark2223 күн бұрын
Great expanation of the inclusion/exclusion proncople
@elreturner12277 күн бұрын
e and pi are the most interesting numbers like “oh you have a weird value for this problem which no field of mathematics even comes close to?” Plug in e or 1/e or e^2 or the eth root of e or e^pi just keep plugging variations of e and pi and it’ll probably work and if it doesn’t even Euler can’t help you
@daniihh9 күн бұрын
thank you for the pannenkoek2012 music
@TheDavidlloydjones21 күн бұрын
As is obvious from their accurate, lifelike portraits, the men in this diagram are misidentified. They are actually Man 47, Man 312, and Man 14,703. 1. 2. and 3 all died in the pneumonia epidemic of 1919.
@mattadams22927 күн бұрын
At about the 18:55 mark, when reindexing to start at i=0 instead of i=2, wouldn't this change the ending index to (n-2) rather than n? Once we're considering the limit as n goes to infinity, this change no longer matters, so the 1/e result is unchanged.
@GayAnnabeth6 күн бұрын
oh hey, alternating harmonic series, love to see it
@nullmeasure615517 күн бұрын
Hot example, honestly. This is also the most lucid presentation of inclusion-exclusion I have ever seen. Well done.
@WrathofMath16 күн бұрын
Thanks a lot!
@orsoncart80219 күн бұрын
That reminds me, not of hats but of doggies’ Rsoles: The doggies held a meeting, They came from near and far, Some came by motorcycle, Some by motorcar. Each doggy passed the entrance, Each doggy signed the book, Then each unshipped his Rsole And hung it on the hook. One dog was not invited, It sorely raised his ire, He ran into the meeting hall And loudly bellowed, "Fire." It threw them in confusion And without a second look, Each grabbed another's Rsole, From off another hook. And that's the reason why, sir, When walking down the street, And that's the reason why, sir, When doggies chance to meet, And that's the reason why, sir, On land or sea or foam, He will sniff another's Rsole To see if it's his own. In this case e = 1 explains all the sniffing! 😁
@rodrigoqteixeira15 күн бұрын
Nice. You can instead of divising by e multiply by the precomputed 1/e or e^-1
@MitchBurns24 күн бұрын
e is also closely related to pi. e^pi*i=-1. Also if f(x)=f^4(x), f(x) could equal both e^x, or sin(x). Also, since you mentioned rounding, both pi and e round to 3.
@josenobi302224 күн бұрын
Or you know, cos(x) and basically any sine function of the form a*sin(x+b) or a*cos(x)+b*sin(x)
@chronoray878617 күн бұрын
Wonderful video. You indeed really do know how to teach. :)
@Lolly_Bingo6915 күн бұрын
20:05 I was just waiting for it (it was worth it)
@secondbeamship24 күн бұрын
This seems like something that may be non-polynomial.
@tomq649119 күн бұрын
Nice video, well explained. perhaps a more practical application would be that there are several workers assigned a role, you want to reassign their role such that each person has a different role. how many ways are there of doing this. Perhaps role could be changed to position if the application is sport, maybe like the total football of the Dutch team during the 70s.
@J7Handle24 күн бұрын
I can't believe I didn't see 1/e coming. I just thought "ooh, Taylor series for cosh(1) minus Taylor series for sinh(1)". Of course, that _is_ 1/e, isn't it? Reason you have to round n!/e to get !n is that you're only summing the terms of the series up to n, and if you add more terms after that to approach n1?e, you'll only have shrinking, alternating fractional terms starting with n!/(n + 1)!, which at largest will be 1/2, meaning the sum of all fractional terms going to infinity will be strictly between -0.5 and +0.5, meaning you just round to eliminate all the fractional terms, and of course eliminating all the fractional terms gives you all the integer terms, which is the truncated series formula for the subfactorial.
@beansprugget250510 күн бұрын
Very interesting. So basically it goes up to n!/n! and the rest are n!/(n+i)!, I>0, which gets rounded away.
@thatkindcoder75106 күн бұрын
Honestly amazed by how you could confidently go deep into a topic in such an entertaining way. Might be slightly jealous...
@fariesz678620 күн бұрын
this was satisfying bc when saw the approximation before it felt so random yet underwhelming (like what, you just multiply the factorial by an essentially constant factor?) but this explains neatly where it came from
@Sgrunterundt12 күн бұрын
The chance of any person getting their own hat is 1/n. Thus the chance of every one not getting their hat is (1-1/n)^n. This has the well known limit of 1/e as n tends to infinity.
@jamesbond_00722 күн бұрын
Excellent video!!!
@WrathofMath21 күн бұрын
Thank you!
@Jacko_hedgehog24 күн бұрын
This video amazed my mom as I watched this since she never heard of subfactorials before
@flanger00122 күн бұрын
Can we talk about the beautiful Hateno Village music behind this?
@joda76978 күн бұрын
Wait... is this a valid way to define a rounding function? Calculating the subfactorial and factorial exactly, and also saving the inverse of e in memory somewhere? Could one use this somehow to define round() ?
@miezekatze353621 күн бұрын
the moment he said "the probability will surprise you" I thought "probably 1/e or sth like that"...
@Oyakinya-Izuki25 күн бұрын
It's here!
@ollllj21 күн бұрын
do we have a mathertmatical sign for "total electrical resistance of parallel resistors", which is: inverse of all resistances, summed up, and the sum is then inverted again: (resistor1^-1+resistor2^-1...)^-1 This also occurs (identical function) in "exponential smooth-step function" for "smoothing (more than 2) meta-balls or more complex signed-distance-fields" (commutatively), the simplest one one that is commutative and that allows for independent "sharpness" factors.
@julioaurelio21 күн бұрын
That's H(n)/n, where H(n) is the harmonic mean of n numbers. This operation also appears when finding the equivalent capacitance of capacitors in series.
@MURDERPILLOW.7 күн бұрын
1:32 oh god there was a puzzle like this in proffessor layton, you had to figure out how likely it was that 2 people got their hat but one person didnt. I guessed EVERY number from 1-100% only to realise it was 0%
@RadicalCaveman15 күн бұрын
1/e is so hilarious, I'm amazed there isn't a sitcom about it.
@WrathofMath14 күн бұрын
Same!
@SenorGato23723 күн бұрын
You can 1/e if you want to, you can leave notations behind. Because your friends double count and if they double count then they're no friends of mine.
@thesmilingbraniac443824 күн бұрын
12:45 writing (-1)^(j-1) feels illegal
@ZyloSol9924 күн бұрын
I see someone is a fan of zelda and satoral marsh from Xenoblade.
@thetruetri510624 күн бұрын
Hmmm Derangements are actually exactly what I need for my experimentations on creating sudokus. Is there also a way to easily figure out what those derangements are instead of just their amount?
@buddermybacon24 күн бұрын
Latin squares
@thetruetri510624 күн бұрын
@buddermybacon well latin squares only have n derangements not every possible one though
@joshr.678524 күн бұрын
Love the Zelda background music!
@raphaeld927018 күн бұрын
Seems like a useful way to calculate the secret santa arrangements where no-one gets its own gift. Thanks for the video, I wasn't sure if it was a joke video at first, but it was pretty interesting. You earned a sub :D
@danquella33016 күн бұрын
I kept waiting for the hilarity, and was left with the impression that you math types are easily amused!
@Manisphesto24 күн бұрын
I now wanna question what n!n is, is it n! × n, or n × !n...
@callyral24 күн бұрын
parentheses are your friends
@matino082024 күн бұрын
Id say n × !n bc you put numbers before variables like 2x
@mathguy3724 күн бұрын
!n!
@tobithesergal24 күн бұрын
stuff like this is why im not better at math, that’s very ambiguous
@Abdullah-uszZZ24 күн бұрын
well, clearly, the bodmas order wasn't enough 😢😂
@asmithgames592624 күн бұрын
It's always a fun party when e shows up 🤣
@Plasticshavings24 күн бұрын
the real question is why do they care so much about their hats, like should that really the biggest of their concerns?
@deangeloenriquez160323 күн бұрын
They are really expensive hats and they won’t be seeing each other for a long while so it is pretty important to them that they get the right hat
@HopeHarmony24 күн бұрын
you forgot to draw eyebrows on the men
@Domihork7 күн бұрын
The fact that !n is equal to the rounded n!/e is pretty deranged! 11:24 so it wasn't a fever dream! It does exist! I remember doing this in high school but then never saw it again and couldn't remember what it actually does or how it works. And confusingly in my language it would translate to "n over 2" which in English is used for fractions so I never knew how to look for it. I will have a nice sleep today, knowing you helped me solve one years-long mystery.
@SatisfyingWhirlpools23 күн бұрын
My favorite part about it is that the sequence appears more slightly strange (the terms don’t all end in 000…)
@Lore_Guytest21 күн бұрын
So !5 is just 5! Divided by e and then rounded to the nearest whole number? Nice, knowledge acquired
@Lore_Guytest21 күн бұрын
So !5 would be 44?
@WrathofMath21 күн бұрын
Yup!
@WhyCatsCantFly12 күн бұрын
the file select theme in the background makes this so beautiful
@WrathofMath12 күн бұрын
It's a classic!
@captaindapper502013 күн бұрын
"it's the on left now"
@X-boomer19 күн бұрын
Round or truncate?
@AmosNewcombe23 күн бұрын
If we are adding a new operator, what is its associativity? What for example is “!3!”? Is it 2! or !6 ?
@719PsychLgn24 күн бұрын
heres a math hack! start with 0, then *1 + 1, then *2-1, and so on till u reach your value you want to subfactorial
@micknamens865923 күн бұрын
When you start from n=0, the first value is !0=1 (because the empty permutation has no fixpoint) or even via recusion as !0=!(-1)*0 + 1 (=1) [assuming the unknown value of !(-1) is finite], and from there: !1=!0*1 - 1 (=0), !2=!1*2 + 1 (=1), !3=!2*3 - 1 (=2), !4=!3*4 + 1 (=9), !5=!4*5 - 1 (=44), ...
@DontWatchWhileHigh24 күн бұрын
Me: There's no way there's an explicit formula for this Also me after watching the video: Oh right I did this in combinatorics like 7 years ago
@MaxwellClarkeNZX23 күн бұрын
Seems like there's an easier way to count using a recursive definition: let Ai be the set which has *exactly one* fixed point at index i. This means that index i is a fixed point, and the rest is a derangement: !(n-1). Likewise for Bij (the set which has exactly two fixed points). So the number of derangements is going to have the form !n = n! - ((n choose 1) ⋅ !(n-1) + (n choose 2) ⋅ !(n-2) + ... + (n choose n-1) ⋅ !(n-(n-1)) + 1) Then I'm sure we can do some re-arranging.
@mihaleben605124 күн бұрын
Can not comprehend. No formula????
@loomkkoom24 күн бұрын
We had to prove something with derangements without the subfactorial ended up just becoming deranged 😂😂
@xenomancer110 күн бұрын
We can subfactorial if we want to We can leave integers behind Cause your numbers derange And if that seems strange Then you've factorials in mind We can dance We can dance Everybody look at your pants
@FaerieDragonZook19 күн бұрын
If you have a permutation of n elements, where element n is not a fixed point, you can form a permutation with n-1 elements by removing element n and 'sewing' the permutation back together: where f(a) = n and f(n) = b, then sew it by setting f(a) = b. If the original permutation was a derangement, then either the new permutation is a derangement or a = b. If a = b, then the set of n-2 elements where both n and a (=b) are removed is a derangement. Thus, !n = (n-1) × (!(n-1) + !(n-2)). This is contrasted with the equation n! = n × (n-1)!