Binomial theorem combinatorics connection | Algebra II | Khan Academy

[@dumpsterplayer2700]

the best intuitive explanation of the theorem i've ever seen!

[@jds1212121212]

I got to a top 10 university, but I still use your lectures to supplement my studies! You take a non-pedantic approach which is refreshing and actually informative!

[@Crackneet202x]

Do you still use this email

[@Peter_1986]

Yeah. Most teachers tend to be like "yo, I will show some mysterious formulas so you might be able to memorize them, but I refuse to give you any actual intuition". For example, I had to look up the intuition behind sin(a+b) = sin(a)cos(b)+cos(a)sin(b), because the teacher never explained that part, he was just like "use dat formula".

[@bebarshosnie8751]

Laurelindo thats why i study alone and sit my exams when its time i stopped going to school two years ago when i realized it was nothing more than a waste of time right now i study more and get better grades all i need is my books and generous youtube channels like khan academy or computerphile

[@AdityaKumar-op5zc]

@@bebarshosnie8751 same here bro

[@easyphysics5567]

Great man I would like to be like that

[@sean_gruber]

This is EXACTLY the intuition I was looking for! Incredibly helpful. Thank you!

[@johnholme783]

There’s a very simple way of finding the number of a^2 terms. Multiply the a in the first bracket by the a terms in the rest of the bracket’s, then do the same thing with the a term in the second bracket. Keep moving forward until all combinations are found!

[@madmcd5]

I was away the 2 days my teacher taught this lesson and when I came back I was so confused. You helped me so much! Thank you!

[@evelioreyes5145]

Yep, this is definitely going to have made the difference when I show up for my precalculus midterm and all the teacher wants to know is what the first three letters of the alphabet really and truly are.

[@amitkarnam740]

Thanks Sal! Really the intuition needed instead of remembering the formulas.

[@spamdude1]

Most books don't explain the intuition, great job

[@sufsanin1917]

Thank you. I have always been a keen math learner, but I never understood the relationship between binomial expansion and combinatorics. Thank you!

[@23andrelopes]

I just would like to tell you that you are just amazing. I have learned so much with you and your videos. They are really helping me in university!!!!!!!! I believe that you are one of the most important promoters of education in the world

[@TheIdeanator]

...thats a binary sequence of sorts with a & b replacing 1's & 0's. This is mindblowing. Khanacademy math guy, you are awesome!

[@nocturnalvisionmusic]

4:17-4:19 "Intuition doesn't come easy tho" I like that. :)

[@TinyMaths]

Wow, thank you. This even makes understanding the combinatorial proof for the sum of all the coefficients of (a+b)^n = 2^n so much easier to grasp, i.e. 2 possibilities for each bracket (product rule). My Prof stated this but I was left mystified until a few days ago and this just solidifies it for me. Sometimes something so simple just takes an effective explanation like this. Otherwise it can completely muddy the waters.

[@Ahfjo]

@TheGoldenSet Yes. For example, (3 choose 1) is the same as (3 choose 2). Applying (n!)/[(n-k)!(k)!] to (n choose k), the first case will equal: (3!)/[(2!)(1!)] and the second will equal: (3!)/[(1!)(2!)]. Essentially, both are equal to 3, regardless if 2! or 1! switched places in the denominator.

[@far2phat]

I have looked everywhere for this thanks so much - I am going to show it to my further maths class.

[@janicknorman9396]

"Intuition doesn't come easy though"

[@nocturnalvisionmusic]

Love it. :)

[@lifeofphyraprun7601]

You are very nice!Very helpful.The intuition is absolutely crisp!Loved it!

[@asmnasim9054]

you r best brother...their r lot of things I hv never find the right answer from my teacher...its realy amazing how u explain every thing so simply!!! thanks & love from Bangladesh

[@easyphysics5567]

Thank u sir u are awesome. If every concept is explained like this every student will be an Einstein

[@alericp97]

the best teacher online evr!!!!! Thanks!!

[@ScenicWay]

Thank you soo much!!! You are a life saver!

[@sajaldeb329]

😂😂😂😂

[@black_jack_meghav]

You are great bro... I read the same things from the books, thou they told the same things , i couldn't quite get the point. But when you used three different colours , it paid off. Thanks a lot sir ... You are a great person...

[@jquikshot]

THANK YOU SO MUCH, this is the exact question on my combinatorics take home final!!

[@chonzy]

we need more people like you, thank you man, i watched all 3, understood perfectly.

[@zer0ichi]

Aaahh!!! Basically the b is the main here.. this is the fastest way! Thank u!!

[@saroolicious1]

may God bless you! We're going to start this chapter 2mrw n im ready to go! ^^ :)

[@OprahMemorizedPi]

Very well said! You're enlightening.

[@andresnamm982]

The best Thing ever!!!

[@phodilus2141]

we have a saying in here, Turkey that's very appopriate right now "you've been sended to us from god"

[@phodilus2141]

*appropriate

[@zajec11]

true that most of the time we add '-ed' to a word form a past tense of a verb, but there are exceptions (such as right now) where past tense of 'send' is actually 'sent'. :D

[@gabe80million]

capitalize the G and you'll be good lol

[@nineironshore]

I think 3 choose 1 is the only possibility that results in a squared times b. Around 8:00 in the video.

[@GreenDayxRock1]

THANK YOU. You're a life saver

[@rambo1525298]

thanks man, it really helped !

[@nanaafiadwomoh8322]

So here I am watching this years later, and I'm like where are all these teachers?! Why can't they come to my school?!!!!

[@devil1887]

amazing stuff...definitely intuition

[@hughewang1]

wish i had found khanacademy videos sooner

[@AmritGrewal31]

Crazy important aspect of binomial theorem that most books & teachers love to skip over & then they wonder why students find math hard

[@TheToshme]

rely well explain the subject matter i am very satisfied with this lecture.

[@tush16ar]

really a helpfull kit thanks a lot

[@fakespike]

thanks man we really appreciate the time and effort you put into your videos. it helps a lot of us out.

[@MiloshAi]

I totally get it now! :D Thanks :D

[@vak1103]

you are the best Sal...

[@nathanhayagable]

@interiorptr . I already know the shortcut . I searched for it ,well it worked !!

[@Rickrg95]

Combination, It is usually taught together with permutations.

[@middlevoids]

Nice and clean explanation!

[@interiorptr]

this was SO awesome!! I had such an aha moment at about 5:50

[@kieudang4693]

Not as complicated as I thought it would be :D

[@Corruptangel]

It's more of an explanation. And if you want to know what it can be used for ..it can be used to help you understand Binomial Expansion. :)

[@josiedcrazi]

this is great keep it up...

[@Raiyun166]

Thanks a bunch Sal.

[@arjunmax7]

YOU ROCK MAN!!!!!!!

[@eleazararce5716]

Sir, I believe you missed a parenthesis on the second line. I love your videos.

[@tharunishadi3980]

thank you !

[@Vasu-qn6kj]

Bless you, Sal.

[@AmineChM21]

what a boss you are

[@bilalaamer30]

Can't we write it as 3 choose 3 at the beginning if we had to choose, from 3 things, the terms in which only 'a' is multiplied?

[@swiftlaker6]

Brilliant

[@lonersllc7456]

THANK YOU THANK YOU THANK YOU

[@Rexie021]

thanks~

[@momomtjiddu3920]

you gave me a glancing intuition

[@Peter_1986]

Sal is obsessed with colours. 8-)

[@SuperHajr]

wow it is quite good! i dnt knw what the hell my teacher ws doing in class but nw i kinda understand, THANKS ^___^

[@interiorptr]

@nathanhayagable No idea bout ur question, but you can try his website. Go to this same video on his wbsite and you'll be able to ask questions about it and people answer you.

[@Captain_Rhodes]

i spend hours reading a textbook and didnt get it. this video does it in under 12 minutes!

[@a7thd]

very nice video just a small mistake: (3 1) a b² = (3 2) a² b Thanks Sal.

[@starkillerz6235]

No its not a mistake, he is stating that for one he is chosing for a the other for b

[@starkillerz6235]

And tou get the same result

[@starkillerz6235]

Kind of like how (n r) = (n n-r)

[@enjoyablesounds]

yellow Beeee! thank you man.

[@gautham0709]

Whats the continuation for this vid?

[@kdawg2468]

I want to try solving one of these with binary lol

[@ultimatepirate9589]

Ghost Hobo jeeeez

[@eday00]

If math and I were married (and I'd like to think we are these days), you would be our marriage counselor. You help me love math.

[@oel97]

Thank You Genghis Khan!

[@alakazam7896]

not gonna lie, you didnt have to make me suffer that 5 minutes you could have just skipped to the end?

[@Hotenjin]

who ever disliked this, u could inquiry for an explanation instead of disliking this.

[@molkhal]

never seen so much positive rated comments

[@Lieaonida]

yellow b,b! green a,a! blue a,a! plus yellow b!

[@khantabish2410]

You smoked it!

[@black_jack_meghav]

# _GIVE_ _THIS_ _MAN_ _A_ _NOBEL_ _PRIZE_

[@AdmatHakodesh]

4 math professors just love their white chalks...

[@PROFIEMMI]

all I do is cram dis , I think now i understand it.

[@PROFIEMMI]

normally , most presentation bore me and i end skipping it, but dis one kept me down , trust me I saw the three of them

[@noonesherem8782]

Finally understand after a year and a half thank you

[@MelissaSusan1997]

What I don't understand is: Combinations don't take into account order. They only show 1 number for aab aba baa. Why does it come out right? You would think you would need the formula for permutations. Can anyone help me on this?

[@ShakkIsLife]

In the product, (a+b)(a+b)(a+b), the 6 terms are actually distinct (which explains the colour coding) and thus, combinations are involved. If the product included subscripts, (a1 + b1)(a2 + b2)(a3 + b3), where algebraically a1*a2*a3 = a^3, but conceptually a1, a2, a3 are different elements in a set, then the coefficient of the a^2b term is represented by selecting 2 a's from the set {a1, a2, a3} where order is obviously irrelevant.

[@ultimatepirate9589]

MelissaSusan1997 he denoted them with colour so that nobody asks this question XD

[@andresid]

OH MY GOD YES

[@pratyushpathania7030]

❤

[@kaushikdr]

Why is it combinations and not permutations? Don't you want aab, aba, baa?

[@11matt555]

Whats all this "3 choose 1" stuff? My teacher never even mentioned anything choosing anything. o.o

[@kunwarsingh5227]

gr8

[@twixter0]

am i the only one that gets easily lost in math? :c

[@bebarshosnie8751]

if it wasn't gay i would've married sal his videos never fail to clear everything up

[@shifterdude]

substitute for my math teacher please!

[@37icky]

Was I the only one who shouted GREEN B :3

[@12345papad]

Yes, Allahu akbar

[@nathanhayagable]

sal , if i were given (x+y)^100 , and i was ask for the sum of coeffecients ! can you give me a short cut !! it would take years to solve this crap !

[@SquirrelTrapper]

why are people amazed by these videos? it's basic math... try tackling differential manifolds...

[@lmsnews7581]

Jesus Christ, 240p quality? Man quality sucked back then...

[@sparkgin]

!!!!!

[@darul-hikmatv4981]

Nie e

[@darul-hikmatv4981]

That's nice thanks very much for That

[@aqp93]

out of 183 viewers, 1 failed its exam

[@Number1InBlackOps]

huh?!

[@ultimatepirate9589]

math was awesome then binomial expansion happened and my brain got raped

[@ultimatepirate9589]

then khan saved my life AGAIN

[@badshabz1]

sal i think i love you.....

[@pjensen1123]

bloobie