離散數學CH6、CH8

Welcome back, everybody. Today, we're going to be diving into some lecture notes and recordings that you sent in. How cool. Yeah. And it's all about- Combinatorics. Yeah, combinatorics. Cool. Basically, it's a math of counting and- Possibilities. Possibilities and arrangements. And selections. And selections. Yeah. And I know that might sound a bit dry, but- Yeah, it sounds super dry. But it's actually everywhere. Oh, it really is. Yeah. Yeah. Like, if you think about, like, calculating the odds of winning the lottery or, like, the kind of cryptography that keeps your online transactions safe, that's- It's all combinatorics. It's all combinatorics working in the background. Yeah. And the cool thing is, is that most of the source material was actually in Chinese this time. Wow. Yeah. That's awesome. So we've got it all translated for you, though. Great. So- Are you ready? Are you ready to have your mind bent a little bit? Yeah. I think so. We'll even touch on how, like, dividing money into denominations, it's kind of like arranging stars and bars. That sounds crazy. I know. But it's true. It's true. One of the most powerful tools in combinatorics is something called the pigeonhole principle. Oh. Okay. The pigeonhole principle. It's deceptively simple. Okay. Let's unpack that a little bit. Okay. Sure. The pigeonhole principle. Sounds pretty straightforward. It does sound pretty straightforward. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. 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It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. It is. Let's say you had an example about distributing objects into containers. Yeah. Let's say you have 17 identical objects. Yeah. And you want to put them into three distinct containers. Okay. A generating function lets you represent all the possible ways to do that within a single function. So by looking at that function, we can figure out how many ways there are to distribute those objects. Exactly. Okay. And generating functions are even more versatile than that. Really? You can use them to derive other formulas. Oh, cool. Like the one for combinations. Oh, wow. Okay. Yeah. That's impressive. We're really diving into like the nitty-gritty of combinatorics now. We are. Yeah. What else is there? Well, we've got the inclusion-exclusion principle waiting for us. The inclusion-exclusion principle. It's all about accurately counting elements and sets. Okay. Especially when there's overlap. Like imagine you're trying to figure out how many students are majoring in either math or computer science. Yes. You can't just add the two majors together. Right. You'd be counting those double majors twice. Exactly. Yeah. The inclusion-exclusion principle gives us a way to correct for that overlap. So we include everything. Yeah. And then exclude the things that we've double counted. Exactly. Okay. It's a crucial principle not just in combinatorics, but also in probability, computer science, and even social sciences. Wow. Okay. Yeah. Remember that example from the notes about students majoring in both math and computer science? Yeah. Yeah. You can't just add a number of math majors and computer science majors because you'd end up counting some students twice. Right. Exactly. Yeah. The inclusion-exclusion principle gives us a way to correct for that overlap. Right. We include everything. Yeah. Then exclude the things we've double counted. Got it. Okay. It's a simple idea. Yeah. But it has powerful applications. Yeah. The notes actually used it to calculate something called onto functions. Onto functions. Okay. Think of it like a matchmaking service. Right. Everyone gets a match. Okay. That's an onto function. Okay. Everyone in the second set is paired with at least one person in the first set. Okay. Gotcha. Yeah. Okay. Last but not least, we have to talk about derangements. I remember that from the source material. Yeah. Like that example about mixing up envelopes. Yeah. It's just such a relatable- It is. It's a chaotic scenario. It's a perfect illustration of a derangement. Yeah. Everything's- Everything's out of place. Out of place. In the wrong spot. In the wrong spot. Yeah. And- And we actually have a formula- Oh, really? To calculate the number of derangements for any given set of things. Okay. So there's a formula for that chaos. There is. Yeah. It uses factorials. Okay. And alternating plus and minus signs. Okay. Yeah. I vaguely remember that from my math classes back in the day. You're right. The formula looks a bit intimidating. Yeah. It's basically a way of systematically excluding all the permutations where at least one thing IS in its correct position. Okay. So we're subtracting out all the non-derangements. Yeah. To get to like the pure- The true number. Yeah. The true number of arrangements. Yeah. It's like we're refining- Our count until we get to just like- Only the pure chaos. Yeah. Yeah. Yeah. Okay. Cool. So it sounds like we've covered a lot of ground already from those simple pigeonholes. We have. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. 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