Combination formula | Probability and combinatorics | Probability and Statistics | Khan Academy

Courses on Khan Academy are always 100% free. Start practicing-and saving your progress-now: www.khanacadem...
Watch the next lesson: www.khanacadem...
Watch the next lesson: www.khanacadem...
Missed the previous lesson?
www.khanacadem...
Probability and statistics on Khan Academy: We dare you to go through a day in which you never consider or use probability. Did you check the weather forecast? Busted! Did you decide to go through the drive through lane vs walk in? Busted again! We are constantly creating hypotheses, making predictions, testing, and analyzing. Our lives are full of probabilities! Statistics is related to probability because much of the data we use when determining probable outcomes comes from our understanding of statistics. In these tutorials, we will cover a range of topics, some which include: independent events, dependent probability, combinatorics, hypothesis testing, descriptive statistics, random variables, probability distributions, regression, and inferential statistics. So buckle up and hop on for a wild ride. We bet you're going to be challenged AND love it!
About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.
For free. For everyone. Forever. #YouCanLearnAnything
Subscribe to KhanAcademy’s Probability and Statistics channel:
/ @khanacademyprobabilit...
Subscribe to KhanAcademy: www.youtube.co...

[@JoelChristophel]

8:26 "Memorizing is a good way to not understand what is going on." Nailed it.

[@scitwi9164]

And the best way to not remember it after a month too :q

[@haridaskr1586]

I know..

[@AlexandrBorschchev]

Yes, there's been huge bias against memorizing. Asians did memorize a whole lot of concept around formulas, and mathematicians also memorized equations and soon understood them by studying it themselves in their careers. It's important to note that memorization is not directly a way to understand concepts, rather it relies on the intuition and base understanding to retain information and use it to purpose for later use. I've had people who memorized ideas and hold on them through the higher levels and they rely on it to apply to new formulas for solutions. It sometimes depends on perspective and definition of what memorizing really is, it could be a form of indoctrination and brainwashing to some, and to others there are more ideas that are gained by memorizing. It depends on the situation at which memorizing suits best, but it also depends on the mindset, which is a bit more complicated and expands about philosophy and history.

[@teamholloway7865]

@@AlexandrBorschchev Can you elaborate more on the mindset phase?

[@cianscheer655]

@le mencho are you hearing yourself? you named 3 white scholars and then proceeded to use that as a basis to say "[asians] are maybe good at learning, but not in pushing boundaries". Idk why ethnicity is in this thread to begin with, but what the heck is this comment?

[@NestyPeach]

I got it! Thank you (I didnt even cry this time!!!)

[@mikejonesnoreally]

That made me laugh a bit more than I'm comfortable with..... XD

[@shafiyanasim5381]

@@mikejonesnoreally Exactly XDXD

[@PenandPaperScience]

Awesome! If you want some more practice on such problems, I made a video where I solve them step-by-step. I might help others out there (since your reply is 2 years old :D )

[@matthewtanaya1468]

Wait why am i learning this in 10th grade

[@damian4091]

@@matthewtanaya1468 Cuz you can take statistics in 11th (like I currently am)

[@LisaHall1014]

I'm currently studying for my GED and it feels hopeless. But people like you are helping me to understand this stuff and gives me hope that I WILL pass that test. THANK YOU so freaking much!! :D

[@jasonmorris9330]

Hope you passed, but this shouldn't be on the GED test of any state. This is 14 year (college sophomore) math.

[@wilfy8545]

@@jasonmorris9330 what is the education system in the us? i have to do this for my GCSEs in year 11

[@jasonmorris9330]

@@wilfy8545 This is usually taught in Statistics and General Math, in college. It can be taught as part of high school Algebra, but just basic probability.

[@aquafinner1505]

@@jasonmorris9330i’m not in america but it’s in the ged here !! i’m not sure about the ged math but it comes up really often on the ged science it’s sad

[@bestbotreview]

YOU KHAN DO IT!!!!

[@jaylanka2037]

:)

[@ericdouglaserbe3356]

All night long

[@jawaun1875]

eXo HellHound 😂

[@Fun_maths]

Why?

[@nocturnalvisionmusic]

@@ericdouglaserbe3356 :)

[@yakunsharma4891]

The day I get a job - my first donation will be to Khan Academy. I cannot appreciate you enough!

[@lilwoodiewood3457]

so u got that job yet

[@Ackansha]

@@lilwoodiewood3457I am curious as well 😂

[@RennesPuffs10]

HI DID U GET THE JOB YETTT?

[@kenniemozey2211]

You got job already??

[@skilz8098]

I haven't had a probability or statistics course in almost 25 years. This was a great refresher. When it comes to mathematics I don't forget too much, but this was one of those things that would allude me since I don't use it all too often. It's good to be reminded at times and this was a very simple and elegant explanation for basic Combinatorics. It also shows the power of using factorials along with its relationship to the binomial coefficient.

[@andrewsmith3613]

Excellent explanations. I have a degree in economics and the highest UK high school maths qualifications, but I never really "got" combinations and permutations until now. I'd just learnt the formulas by rote, but now I really understand. Onwards to a MSc in Statistics in a year!

[@123qwe321ify]

And as Moses spoke these words at 10:30 the lord opened up a path through the fog in my brain with a strong east wind. The wind blew all that night, leaving only clarity.

[@a1x45h]

4 years later the exact thing happened to me I hope you are doing fine fellow person.

[@estherkammu1015]

lmaoooo

[@leahjaelgreen]

I absolutely love your video, I have been struggling with this. You actually made it make sense to me. i’ve been taking an online class, so I’m teaching it to myself. Please don’t quit making videos I’m coming back to you! Thank you 🙏

[@monoinluv4894]

I have an assignment over this topic and you're making it much easier to understand than lecture and discussion!

[@juanmoralesdracus]

Let's hope this complements your work (amazing, by the way). This helped me join both ideas: - Permutations: if we have 6 elements (A, B, C, D, E, F), picked in groups of 4, we can get 6*5*4*3 = 360 different groups of them (permutations formula, already demonstrated by you), which look like ABCD, ABCE, BCFE, ... - Combinations: from those 360 permutations, how many of them are included in the same combination? In combinatorial world order of elements doesn't matter, so ABCD = ACBD = ADCB = ... So, in other words, if we now pick 4 elements (in this case, A, B, C and D, because they form ABCD, which is the group we are studying as an example), in groups of 4, how many permutations of them can we get? Since this is a permutation and we know the formula (4 elements in groups of 4), we know we get 4*3*2*1 = 4! = 24 permutations. This means ABCD = ACBD = ... for a total of 24 permutations which are inside the same combination (A, B, C and D in whatever order). So, if 24 permutations = 1 combination, we see that nº permutations = nº combinations * 24, then nº combinations = nº permutations/24 = 360/24 = 15 combinations. This, of course, also works for another 4 letters (BCDE = BDCE = ... 24 permutations for 1 combination of letters B, C, D and E, and so on). Recalling: if we have 360 permutations of our elements and we can make groups of 24 of them which belong to the same combination, then we only have 360/24 = 15 combinations Generalizing: For n elements, taken in groups of k elements, there are n!/(n-k)! permutations. Now we take k elements (which make a group in the previous step), and take them in groups of k elements, to have k!/(k-k)!= k!/0! = k! permutations that belong to the same combination. Finally, we divide n!/(n-k)! permutations by k! of them which are grouped in each combination and have n!/[k!(n-k)!) combinations of n elements taken in groups of k elements. I hope this serves you all, for me it has clarified everything finally. Thanks!

[@zhaofengzheng2923]

Why am I paying thousands of dollars tuition fee?

[@inquisitorbaldy]

Because you live in a barbaric country where education isn't free?

[@katherinemays9856]

I'm not - I go to University of the People and pay very little but I'm still enjoying my education and the benefits of exploring mathematics. Khan Academy is absolutely wonderful because he seems to remember his learning experiences and teaches you about those mistakes and how to avoid them. He does this instead of simply teaching the material presented from a textbook in best-case scenarios without the added flaws that students will inevitably make. He does this consistently through all of his videos on a wide range of subjects and that's why I love Khan Academy.

[@sneakybeaver8866]

thaaaaat part

[@117swordsmen]

So you can have a piece of paper that says you're allowed to earn money.

[@mikejonesnoreally]

For the reason that there is no knowledge that is not power but without that little piece of paper that you get for attendance, you're like a battery with nothing to fuel. It's not impossible to do something of meaning, but wow, *wow* is it a *lot* harder! =3

[@Cheva-aree]

OMG, i understand it already... This is much better than listening to my teacher's nonsense talking... By the way, im gonna make this into a report//' "D

[@MCtechh]

Same is happening with me!

[@alejandrom8766]

ay!, cochino

[@kyandrew7942]

I have an exam in 10 minutes and I understood a lot more from this compared to reading the textbook

How'd the exam go?

[@kyandrew7942]

@ Well I passed with a 79% Grade and I was surprised to realise I learn faster through yt vids

@@kyandrew7942 congratulations mate, that's an excellent grade altogether you should be massively proud.

[@ebukaijezie3595]

​@ was about to ask this lol

[@karepainting4929]

​@@kyandrew7942 At least you didn't fail thanks to this video 😊

[@Snoo29293]

I actually understood this, I am not studying probability or statistics I was actually searching for some other algebra stuff but I found the concept interesting and now I know how to calculate combinations which is cool

[@chengli7046]

After months of struggling, this is the clearest and easiest explanation of permutations and combinations I've come across. Thank you! You are truly one of the greats of our time

[@JustinLeatherwood]

I’ve been hitting my head against a lot of these concepts, the first 3 minutes, that visualization finally made that formula click, thanks!

[@oddyellow6943]

I was really stressed because of permutation and combination. Never would've thought that it's not that hard to understand. Thank you for giving us a great explaination.

[@swfsql]

C = nPk / kPk = (ways to select and arrange items on the places) / (ways to arrange items on places | (given a) fixed selection of items) = (ways to select items to the places)

[@caht6468]

guy has a soothing voice, we need all math teachers to be like him

[@tuomokriikkula963]

Khan Academy is smart for hiring fluent English speakers. This is easy to follow all across the globe.

[@tuomokriikkula963]

The teacher in this video is also remarkable.

[@devkathuria2842]

if u didn't get it , then start listening from 10:30 again and think Peace.

[@safiulfaiyaz7038]

The K factorial with which we're diving is better called the "Number of ways we can arrange the SAME 4 people" With this, we can only get the number of ways we can arrange the 6 people in 4 chairs with each arrangement having at least one different component, so all the scenarios where we arrange the same 4 people in different order are counted as one as we're diving the no of permutations with the number of ways we can arrange the same 4 people.

[@johnjordan3552]

I didn't understand where this things is coming for. I looked for books about it but there was even more advanced stuff that I can't understand but now, thanks to you I understand where it's coming from. Thank you, for all

[@MrJDog3000]

Thank you so much. Studying for the GRE has slowly been killing me and this one video made a world of difference. Thank you again.

[@silverphoenix_1756]

8:35 "Because memorizing is a good way to not understand what is actually going on." Loved that :')

[@sevenaries]

8:35 School System: *shocked Pikachu face*

[@ayeshashaikh4433]

I seriously don't know what I would do without you, thanks a lot!

[@Justin-c4h]

That was helpful, thanks. I just wrote a Python program that uses this formula. The only thing I had to adjust for is when n = k, having a 0 in the denominator causes problems... of course the answer is just 1 in this case, but programs needs to account for things like this.

[@underaglassmoon6056]

If anyone is still confused, here is another way to look at it: imagine a pool, this pool has 120 balls. Let a ball represent a single permutation. So in our pool, we have 120 permutations (120 ways to choose things where the order does no matter). Now let's keep our goal in mind: we want the number of ways to choose things were order does not matter, so intuitively a number smaller than 120. Back to our concrete example: we have 120 different permutations in the pool. What we want to do is group those permutations into families of objects (eliminate the duplicates). Let's say what we wanted to do is choose 3 of the letters ABCDEF,such that order does not matter. Consider one of the families to be the ABC family:(ABC,ACB,BCA,BAC,CAB,CBA). As Sal said, the size of a family corresponds to k! or 3! (6) here (I chose 3 instead of 4 to keep it short). So if each ''family'' has a size of 6 (which corresponds to 6 different permutations or 6 balls), and there are 120 total permutations (120 balls) in our pool, then we can group our 120 balls into: 120balls/(6balls/1family) = 20 families. So there are 20 families (each family is a combination so 20 combinations).

[@lilwoodiewood3457]

the answer is just 120 divided by 6 which is 20 so whats the point of making it so much more complicated in that context ... their is none its pointless now it snot entirely pointless in other situations but in your example that can just solved wih simple divison so whats the point of adding extra steps when elementARY school level math can get it done more advenced group combination math has value like say if i were to have 600,000 items in 6 sub groups and i wanted to make as many unique groups as possible (to be unqiue a group it just needs one unique item but one item of each subgroup needs to be represented in each group each sub group has a equal number of items items can repeat as long as each group has atleast one unique item like say i have abcdef abcdeg abcdeh abcdea abcdeb etc. ) now thats a example that actually needs group combination math it cant be solved with simple divison

[@ImTheCoolest-op6dp]

This is an amazing video. I think I will rewatch it tomorrow when I’m less tired but it is just what I’ve been looking for. Other videos just expect me to memorise but you actually tell me what is going on so I know I’m not learning nonsense.

[@sweetmela1835]

I'm jumping!!!!! I finally got it!!!!😭❤❤❤ thank you so much!!!!

[@the1onlyc77]

Thank youuuuuuu!!!!! I got it!!!! I think it’s your voice!!! It’s sooo soothing and it makes me feel like we did it together!!!! Whooo hoooo!!!🙌🏽.. I swear I was sitting at this table in tears before I clicked on here.🥰🥰

[@typeterson8376]

if it makes a difference, to do this in Python or MATLAB, you can find a factorial of a number in MATLAB by doing 'factorial(n)' and a factorial in Python by importing the math module and doing 'math.factorial(n)'. Similarly, you can do the n Choose k method in MATLAB by doing 'nchoosek(n, k)', and you can do it in python by importing the math module and doing 'math.comb(n, k)'. Automating these probability and statistics things is so much easier and quick to learn.

[@bikerduckd]

Last few sentences made it super easy to understand! Thanks!

[@rpuraaaaab]

9:43 is the moment I got enlightened! You're a lifesaver!!!

[@dansierrasam79]

Thank you for explaining it very well. Appreciate it greatly!

[@hari7591]

yes! that makes total sense!! thank you!!!

[@DaniyalAsifKhan]

THANK YOU KHAN SAHAB...

[@paradox6647]

This is the greatest explanation I’ve ever seen for this concept, easy to follow and understand.

[@sebastianohomberger8340]

I've watched all your videos about permutations and combination and I found them extremely useful and helpful. I have a question: you separated permutations into 2 groups, 1 where you can repete things in the permutations and 1 where you can not, isn't this division present in combinations too?

[@aditijain6445]

8 years old but still the best videos on the internet

[@ericfloydbaliquig5813]

Thanks for explaining these concepts clearly. I wish this educational videos KZbinr will last forever.

[@geraldhoxha]

What if there are k=9 seats and n=3 people? All the examples I saw on KZbin tells that the n (item) is greater than k (hole for example). I make a thought experiment: Let's say we have a chessboard (10x10) and there are 3 items on it, how many combinations there are for all the items on the chessboard? I'm so confused. Using the formula that is shown in the video, and swap kn in the formula, there are 504 combinations, which I think there are a lot more than that. If we don't swap kn, then it's undefined, because the factorial is defined only for positive integers and 0 (0! = ).

[@philamras3732]

Brilliant explanation. I never fully understood where this formula came from but now I do.

[@s1n01891376]

What an example! Using this you can come up with the formula on your own each time

[@bhupendrabissa2883]

Mario, Your are just great. It was so simple that I have started wishing, If, I would have you as my Math's teacher I would have got job in NASA.....Great Work.

[@ashishkumarahuja212]

Nice way of teaching :) For GK session.. I found Vidya Guru videos really helpful, all their video have excellently helped me in exams.

[@SY-me5rk]

this will make a lot more sense after watching sals videos on permutation. how do teachers even manage to make something so simple so hard.

[@faqihmualim6290]

So long time i have memorized this formula, but now understand the proof of this. Thank You

[@hklausen]

I was able to caculate the combinations before, but now I also understand the algebra :-) Thanks

[@larzperson9601]

thank you, never understood why the formula was the way it is. And now I can see why it is the way it is.

[@SBskagod]

it helped me to think of it like this: each 4 letter combo is represented in 24 (=4!) unique orders within the 360 possible orders you can make using 4 letters out of 6 optional letters. picture all of the 4-letter-combo's different orders next to each other... ABCD, ACBD, ADBC.... There would be 24 orders of those 4 letters. then, next in line is the 24 orders of the letters ABCE. Then 24 orders of the letters ABCF, and so on. if you circled each group of unique 4-letter-combos, putting all 24 orders of those letters into one circle, you would have 15 circles.. it's the same thing as dividing 360 by 24.

[@melvin4441]

These are great ideas for beginner computer programs too if you're struggling to think of an idea.

[@alvin806]

I think you need to state the problem first before you proceed to the solution. In some way give an example problem (in word problem).

[@ilse8239]

Wooowwww, why can't my teacher explain like this hahaha, so simple, my teacher makes it seem more difficult than it really is. Great explanation, very helpful. Thanks so much! :)

[@sangeethashanmugam6079]

I referred so many materials, and finally got clarity here!! Thank you so much!

[@richardfredlund3802]

thank you... I know this already but like you say it can be confusing, and this was exactly the clarity I was looking for.

[@amondevecasones1929]

One of the greatest teachers/tutors to be born...

[@crunchy_poptart3979]

I really like how closely this relates to the pascal triangle ! Very nice

[@沐屿-y9j]

YOU ARE BEING REALLY HELPFUL！！！Excellent explanation, trying to really understand these two formula always be confused for me before I saw this video.

[@pokecraftians2374]

The last part of the explanation was the best! I literally just understood everything after that explination

[@thurainhtetko]

Thanks this channel is so useful. Whenever I don't understand something, searching in the youtube i see your video that explain me.

[@sharadmitra4318]

As always, clear and coherent.

[@airicadaria1739]

You're the best! Thank you so much.

[@KoalaStudy]

Amazing tutorial! I'm learning this at the age of 12 but I can fully understand your steps and the chapter :)

[@hailaconti9007]

from brazil 🇧🇷 and that was the best class that i found, thank u so much!!!!

[@josephv7762]

This helped so much. Bless you.

[@nocturnalvisionmusic]

That's great :)

[@varunprathap8359]

you always make it simple to understand, and also another way to view the nCr, is nCr = (nPr)/(rPr)

[@moisesm9602]

The way I got so happy when i solved my math problem after this video

[@haaey1197]

Lol, “In videos after this i’ll use other example rather than sitting chairs” 😂😂

[@BushcraftAI]

Thanks a ton! You helped me my algorithm class!

[@qeithwreid7745]

The way the proof is going becomes fairly clear at about 1 minute. This is to your credit not mine 😁 A well constructed lesson. Thank you.

[@qnu1909]

Im learning math for fun and this is great ! Ive longed to see a clear explanation of this.

[@jack-mn7xe]

Great video, I'm so clear now

[@lraoux]

Thank you. I needed this.

[@theromancobra]

Khan academy I LOVE YOU!!! You got me through my ME degree nad youre going to get me through my MBA

[@Zenfullvnn]

my teacher is killing me with this permutation. omg and now we're learning about conditional probabilit. I'm lost

[@SANKETJOHN]

This was cure of an old sickness. Thank you.

[@nicoong8944]

great video with a great explanation! Understanding the formula is easier for a short memory person like me actually :)

[@welt618]

best math teacher along with dont memorise

[@jonahkane7027]

God bless you and keep you sal. Thank you thank you thank you you have saved me from an F

[@sirajtayyabkhan6820]

Man I love the way you make us vivid!

[@aviwemayengeshe7569]

Thank you, I hope I am going to pass the test with the understanding of this video

[@justis3277]

aviwe mayengeshe. Uhh. uhh. H. uhh. H uhh. uhh. uhh. uhh. uhh. uhh. H. H. uhh. H. uhh. uhh. uhh. uhh. H. uhh. uhh. uhh. H. uhh. uhh. uhh. uhh. H uhh. hubbub. H. uhh. uhh. H. H. H. uhh. uhh. H. uhh. uhh. uhh. uhh. uhh. uhh. uhh. uhh. uhh. uhh. uhh. uhh. uhh. uhh. uhh. uhh. Uhh. uhh. uhh. uhh. Highbury. uhh. uhh. uhh. uhh. uhh. hubbub. uhh. Uhh. H. uhh. uhh. H h. H. H. H. uhh. H. uhh. uhh. uhh. Highbury. uhh. uhh. uhh. H. uhh. Highbury. uhh. uhh. uhh. uhh. H. uhh. uhh. uhh. uhh. H. uhh. uhh. uhh. uhh. uhh. uhh. uhh. uhh. uhh. hubbub. uhh. hubbub. uhh. hub. uhh. Highbury. uhh. uhh. uhh hubbub. . H uhh. uhh. uhh. Highbury. uhh. bub. uhh. uhh. uhh. Highbury. uhh. uhh. uhh. uhh. uhh. Highbury. hubbub. Hubbub. uhh. uhh. uhh.

[@sanan22]

in the example of this video. how many different combinations will there be if chairs could stay empty or be shared? is the answer (7^4)/4! or (13^4)/4! or (12^4)/4! or ((6+6C2+6C3+6C4+6C5+1)*4))/4! or something else?

[@kawtarbouzaidi53]

Thank you sir !

[@munoxasan7123]

Thnks for education videos special math

[@Optimistical]

Thank you guys so much. Combinations was one of the subjects I couldn't figure out.

[@harishkumar9733]

A big thank you mate. I went in repeat mode to get this understand. I cried once i got it.

[@Lokkii59883]

Thanks for helping me for my Math Test next week!!!!! Your awsome

[@shivamgupta1719]

Thanks you for making things simple love from India 🇮🇳

[@rosiemurch7345]

I’m 13 and they have explained in 11 minutes what my maths teacher couldn’t in 3 hours of online lessons

[@thanksforyourmoney5083]

Moment of clarity 4:50-5:16

[@mathysicssaransh7939]

Thanks a lot!!!

[@elaphahmed5448]

you are such a hero ..thank you so much

[@godlybench7581]

I have become enlightened

[@findmywellbeing537]

You're awesome. I now love this channel so much. A real life saver!

[@zeeshanhaider148]

Thank u.. Now i dont have to remember the formula.. 👍👍

[@ntagwankokisabanzira6578]

WONDERFUL!!!

[@estherkammu1015]

you da best sir