JEE ADVANCED 2026 || Paper 1 || Question 11 || Combinatorics!

追加: 2026-05-30

[00:00:00]All right. Hell yeah. So, we're going to do another J advanced 2026 problem. Now, we're not going to complete this. We're just going to do like the introduction to a problem. And so, hell yeah. So, let's read it. It says the number of ways to distribute 10 identical red pens and 14 identical blue pens. So, right off the bat, we're going to have 24 pens, and we're going to distribute these among four persons so that each person gets six pens. So, after you distributed them, you always have six pens um going to four people. So, you're always going to have four people and each person is going to have six pens.

[00:00:48]Now, I did look up the solution. I have no idea what the answer is. You know what I mean? the number. I don't really care about that. Um, but they did say one of the clues was that if you figure out the different ways to distribute the red pens, the blue pens have to kind of fall into place. Um, but what this video is going to be about, and one thing I had to look up, I had to stop this problem because, um, I I kind of understood that if you had identical pens, um, you know, here's a red pen and then and then here's a red pen, is that you can't distinguish between them. And so if you got this red pen, you wouldn't be like, "Oh, no, fair. I wanted that one.

[00:01:40]And so I think it's useful for this especially when you're divvying up just random items to be like, "Hell yeah."

[00:01:47]Let's say you're um divvying up old, you know what I mean? I was I was going to make it like uh um you know what I mean?

[00:01:58]I don't know. I I was going to go with women or something, you know what I mean? Or like dates, you know what I mean? Hey, this is Hey. Yeah. Like the dating game. Hey, one, two, or three.

[00:02:08]Now, you know, you got some now, now you got some uh stake in it and stuff like that. Um, you know what I mean? Uh, the three women aren't identical or whatever, you know, for for that uh matter. Um, the other thing that was confusing to me was like, is the people are the people identical? And no, so for this problem, you know what I mean? you have um four very uh different people and um and so now for this I thought like hey these people you know have they need to care about they actually do care about what choices they got and I thought that was very useful going into a problem like that because who the hell cares if this person first round got four red pens and two blue pens. There's no there's no skin in the game, you know, compared to, you know, let's say that person got what they wanted, you know, and this person didn't get what they want. Now there's skin in the game.

[00:03:16]Um, and so that was very useful for this problem to show that these people are not identical.

[00:03:23]That was the biggest traffic jam reading this for the first time is just like, well, if you have identical pens, are the people identical? Do they really care? You know what I mean? Like, um, but yeah, so each of these people can get, you know what I mean? Um, yeah. So if you have if you have like uh yeah these two red pens and each person, you know, either gets none, those two get none and these two get these two red pens. Um and let's say one's blue. So one's red and one's blue. And uh hell yeah. In this situation, those two people are excited. These two people are disappointed.

[00:04:06]But then let's say this person didn't want the blue pen. They wanted the red pen. And this person didn't want the red pen, they wanted the blue pen. So now all of them are disappointed.

[00:04:16]Um so that shows that even from that scenario to okay give this person the blue pen and this person the red pen.

[00:04:30]I'm trying to prove the point that the people are not identical because now this person's like hell yeah. And this person's like hell yeah I wanted the blue pen. And then this person is like, I wanted the red pen. And so there's very different between blue and red compared to red and blue. And so I'm just doubling down on the fact that the people aren't identical in this scenario. Okay. So now I'm going to play around with uh just grouping things. And let's forget the idea that we have two different color pens. Let's just say we have one red pen and then just one person. Cool. It looks like there's one way to distribute the red pen to the the person.

[00:05:17]And for this problem, you can say that this person always gets one pen and they're going to get the red pen. Cool.

[00:05:24]So, I think this is useful and I'm going to show you why this is useful. You have the prisoner logic problem that can, you know, I mean, mess with your mind is that you have four prisoners. This one has a hat and they're going to be black or white.

[00:05:44]Um, and then hell yeah. And now you have uh like this tallest back here. These people can't turn around else they get executed or something crazy or they're buried in sand. And um Yeah. So let's let's uh just do this.

[00:06:06]Yeah. And then and then so there's and then the prison guard says there's two black two black hats, two white hats. And at any point and these people are looking this way. They're all looking that way. And at any point, if a prisoner knows the hat on top of their head, they can be free if they know 100% positive the hat on their head. Else they die. And if they don't choose, maybe you actually get life in prison.

[00:06:40]You know what I mean? So, it's like you wouldn't want to choose wrong. That would be, you know, so set the stakes there. Um and for this the most simple thing you can do I think it's like taking a principle like one let's let's add numbers and you could get complicated in adding or just add the most basic 1 + 1 is obviously two um and then even numbers get complicated adding because you can add the opposite 1 + a -3 who knows what that is you know I mean let's say it's your first rodeo and that's just a lot of different things mathematics is your first rodeo.

[00:07:17]You just like, well, I don't have skin in the game. I really don't know what's going on here. Maybe it's more useful to start with one plus a negative one and think of this as I got one. I'm taking away one. Okay, now I have zero instead of going, you know, this way and that way from positive to negative or something. And then so, hell yeah, this is going to be your most basic case 1 plus two. Let's get back to Let's get back to this one in a sec, but let's introduce two people here.

[00:07:48]And then let's make it a little more complicated. We have a red pen and a blue pen. So now, how many different ways? And then each person gets one pen.

[00:07:58]How many different ways can we do this?

[00:07:59]Well, we're going to do red pen, blue pen, or blue pen, red pen. And we've already talked about how the people are not identical. uh they're either excited or they're disappointed or they like red and blue both you know what I mean but that matters and I think we have two choices now I might make a mistake even through this rudimentary process maybe you know what I mean I don't know you know then this is like well couldn't you give both pens to both to one person the other person has zero pens no because even in our problem statement we're saying we have four people they all get six pens so that scenar scenario is out and we're just left with. So, this is super basic. You know what I mean? That one should be totally obvious and I think it's obvious to me. But the fact that I even question that, hey, I'm going to like if I make this even slightly more complicated, I'm bound to make a simple error. And so that's where this problem, the solution is, well, you get to choose the hats. Let's say, you know what I mean? Like you get to choose who you're going to be, which you'd want to be prisoner for, and you get to choose what color hats, you know what I mean? You want to distribute. And so, let's draw this one more time.

[00:09:24]And we got our wall.

[00:09:26]And so, so I'm either thinking of prisoner 4 or I'm just like, let's say I'm the prisoner guard and I want them to have the easiest time. I say there's two black hats and one and two white hats. And I'm going to do this. I'm going to put two black hats here and then a white hat here and a white hat here. And so I already told everyone that there's two black hats, there's two white hats. This person is literally staring at two black hats. It can't be more obvious. And so, isn't that interesting that if this was the most obvious way to look at it, the fact that you just make it a little bit different that now you can't tell that, you know what I mean? The only the only difference between this and this. I mean, you could have a third scenario where um you know, this person has a white hat and this person has a black hat, but still these three scenarios are the only grouping that this person can be well four um cuz this could be black and these two could be white.

[00:10:44]So, I mean, I just chose one where it's so obvious like hell yeah, this person's going to be like my hat is white. Um, here's another scenario. This prisoner is going to be like, "My hat is black because I see two white hats." So, that's the those are the most obvious.

[00:11:00]And so, we're going to use kind of that strategy um for our problem. Um, and I think that's kind of what we're doing. We're taking something that's ah, you know what I mean? And this is so complicated that you have to use the stars and bars approach, which I've never heard of. And even though I know that there's this equation called stars and bars and I looked it up once and I kind of saw the structure, I have no idea what the proof is. Um, haven't looked up any farther than that. I just wanted to experiment.

[00:11:35]And so we're going to spend, you know, another five minutes trying to experiment as hard as we can um on this and then we're going to be done with this video.

[00:11:45]So let's say we have um two people and they get two pens.

[00:11:53]Uh, two pens each. So, there's going to be a total of four pens. And we're going to have two red, two blue. So, now how many combinations? This person could get two red. This person could get two blue.

[00:12:05]This person could get one red, one blue.

[00:12:08]One red, one blue. This person could get two blue. And then this person could get two red. Um, if they both get one red, one blue, one red, one blue, that's the same. This is going to be the same as this person getting one blue and one red and this person getting one blue and one red. The pens are identical. Um, they're in a pile. It doesn't matter. You're going to give them give them the pens simultaneously and so there's no order that you know.

[00:12:42]So there you go. Um, and I think I mean I don't know that's why these are hard.

[00:12:49]Like like I said I I said even here is this the only scenario? I'm not 100%. I feel like 100%. But isn't it weird how there's just a little bit of doubt? This I feel like I'm 90%. Um, but now there's 10% doubt that is just like well is there another combination?

[00:13:07]And so uh I don't think so. Let's move on to something a little bit more. Um, you know, let's have uh let's have six pens, two people.

[00:13:22]So, we got the two people, we got six pens, we got three red, three blue.

[00:13:29]Um, cool. So, we're going to do three red, three blue. Um, and then this person can get two red. Um, one blue, uh, two blue, one red, one red, two blue, one blue, two red.

[00:13:52]Um, zero red, three blue, uh, zero blue, three red.

[00:14:01]And I think so they can get there's um how many different combin I already mess up because this person can get three red and this person can get three red. Okay, I'm just going to double check that scenario because hey, let's say this person loves red. Um hates blue and this person loves blue.

[00:14:27]Hates red.

[00:14:30]So when this person got three blue, very happy. This person got three red, very happy. U but we need the opposite scenario. This person got three red, disappointed. This person got three blue, disappointed. So at least I'm double-checking that scenario.

[00:14:46]Okay. And we have one, two, three, four combinations.

[00:14:52]Um hell yeah.

[00:14:57]And so, yeah, that's interesting.

[00:15:01]This has a ton of combinations. I I I don't see how I'm getting, you know, more um like I I feel like I'm making mistakes because hey, we have four pens, two people. Um let's have let's have three people. I think that'll up the stakes. And then we're going to have we could have um nine nine pens.

[00:15:28]And we're gonna have three red, six blue.

[00:15:33]Um cool. And we could even have four red, five blue. So we have nine pens, three people.

[00:15:43]Um each person gets three pens.

[00:15:49]And so this person could get three red.

[00:15:53]This person can't get four red, but this person could get um two red, one red, they could get zero red.

[00:16:05]Um cool. And if they get three red, they don't get any blue. And so we got zero blue, one blue, two blue, and three blue.

[00:16:23]Um, cool. If this person gets three red, let's go to this person. They need at least one red. Two red, they get um, we could have two red or we could just have one red, one red. So now now we're getting that kind of that uh exponential pattern of just like oh yeah if I have some remainder that remainder could go to this person or it could go to the other two. They're identical. And so this person getting this red and this person getting rid this red is the same as this person getting that red. You know what I mean? Okay. Hell yeah. So we have one red, two blue. And so we're gonna have um we could do three red here or we could do two red, one red um or one red, two red. So even there because we have this surplus of three reds, we have you know um three different combinations. This is zero.

[00:17:26]This is zero.

[00:17:28]And so we have um this was three reds, one red, um two blue, and three blue. So that's a single combination.

[00:17:43]Two red, two red. That's all the reds.

[00:17:46]Zero red. U this must be three blue as well.

[00:17:51]Um two red here. This must get one more blue. And so we're we're double checked on the five blue. Um two red, one red, one red. They need three pens. So this is two blue, two blue. And now we're seeing how if we choose the red, it um it it basically shows you I don't know what the verbiage is, but it shows you what the blue is. If I I'm only going to do this for 20 minutes, so if I run out the if the video runs out, so be it.

[00:18:22]We're going to be done. We could be kind of done right now. I already kind of see this exponential process, but it took till here to actually see that. Oh yeah, we could actually get kind of like some exponential. I don't know if it's exponential. I think it is because that stars and bars uses factorial, which at least 2 * 3 * 4 kind of stuff. That is an exponential process.

[00:18:49]Don't quote me on that term of exponential process, but I've used it in series. This comes up in series a lot in series that involve exponential um equations or summations and stuff like that. Okay, zero red, three blue.

[00:19:05]That means that we have four red. This person can only get three red, but they could get two red, one red, or zero red.

[00:19:13]Um, if they get three red, there's got to be one red here. We already said this person gets zero. Uh, if they get zero, they get three blue. Um, three, one, two, two, three red. And if this is zero red, zero red, we can't have that combination. So, this combination can't exist. Zero red, zero red, because we'd have four red. And that one person can't get four pens. Each person gets three pens.

[00:19:44]Okay, so that scenario and I think we're done. Um, so zero. So we have um zero red, three red, three blue, two red, one blue, one red, two blue. I'm just double checking that each. All right. Hell yeah. I guess we're going to go over 20 minutes, but I just want to show you that there's nine combinations. Uh, and I'm going to point them out. And so we have one combination, two combinations, three combinations, four, five, 6, 7, 8, and nine combinations. And so I can double check this one, but I think the logic feels sound. And again, this strategy is try to pick the most simple and then start to get more complicated.

[00:20:37]You know what I mean? Just like some of these logic problems. And because because this has uh you know the stars and bars which I've never touched. What I have found is that when you're learning new proofs, they feel really technical and they don't feel geometric.

[00:20:57]they don't feel like they have tangible u they don't feel like come from a tangible place. And so I just thought this was a very good start to stuff like this to be like, hey, can I do my own little groupings where I could do it by hand? Like nine combinations wasn't too bad. If this is like 278 combinations, well, yeah, that's going to be madness.

[00:21:20]But now I can at least start the process of being like, hey, even if I was wrong, I could start to set this up and be like, how many combinations are there?

[00:21:33]You know what I mean? Maybe I could get like 40 combinations just kind of in one little scenario and then I have to go to the next one and I feel like I'm going to get another 40 and I'm going to do that 10 times. Well, then I'm going to feel like I'm going to get 400 different combinations. And so here kind of the same thing. Well, I got one scenario, but this one I have like two scenarios.

[00:21:55]And so if I follow that logic, one, two, three, and then one, two, this one is still three. So that's kind of interesting. Um, yeah. And so I would like to spend a little bit more time doing this one by hand, but we're done with this video. uh probably the next video on this one is to start using this ident you know using this problem statement and start doing the process by hand even if it gets tedious even if it feels impossible to complete and even if I make very horrible errors cuz that's the other thing is just like hey do wrong groupings make those errors and see where you are going to make errors just so you can prevent those But it's always always good to make errors and um you know when working with new concepts because you're going to make them at some point and then find ways to be like well why why did I make that error? And so hell yeah. All right we're done with this. Thanks for joining me.

[00:22:59]Booya.