 So lesson one, the fundamental counting principle. Jesse, can I suggest to you that anything in math that has the title with the word fundamental in it means it's really important. It means what we're gonna do is we're gonna take today's lesson and for the next three weeks or so, we're just gonna build on this. In fact, when in doubt, I fall back to today's lesson. This is the one that I'm gonna learn inside and out and then I'm gonna show you shortcuts along the way but if I'm not sure whether the shortcut will work, I fall back to today's lesson. The fundamental counting principle or objective today is to develop the fundamental counting principle. It's really counting without counting. So here's our first example. It says a cafe has a lunch special consisting of an egg or a ham sandwich, E or H, milk, juice or coffee, M, J or C and yogurt or pie for dessert, Y or P. It says one item is chosen from each category, list all the possible meals. Well, listing them would be tough but if I wanted to do it in an organized manner, I would probably use a tree diagram. I would probably go something like this. I would say, okay, I can have an egg or a ham sandwich and after I choose my sandwich, I can have one, two, three beverages, one, two, three beverages, M, J, C, M, J, C and after I've had a beverage, how many choices do I have for a dessert? Two, I could have Y or P, Y or P, Y or P, Y or P, Y or P. What was that? Oh, I thought someone said so much writing. Oh, because that actually is part of the point here. I'm not gonna list them very often because it's too much work. What I'd like to know is how many meals are there but I'd like to try and figure out how many possible meals there are without counting them. Now I can count them right now because if I look at the end of my tree, I have one, two, three, four, five, six, seven, eight, nine, 10, 11, I have 12 meals. There's 12 possibilities. Can anybody see a way we could get the number 12 without listing them based on the information that we were given? Sorry, what? Two sandwiches times three beverages times two desserts is 12 and that's what the fundamental counting principle is. The fundamental counting principle says if you're making choices and you wanna know the total number of choices, multiply how many choices you have for each step. Oh, sorry. As long as none of the choices overlap from category to category. If I had milk up here as a choice, it's a completely different issue as well. Milk, if we're talking cards, for example, that's where sometimes choices overlap, so we'll have to deal with that later. So the cafe features ice cream in 24 flavors. You can order regular sugar or waffle cones. Suppose you order a double cone with two scoops of ice cream. Okay, how many choices of cone do we have? Read the question carefully. How many choices for the type of cone are there? Three, how many choices of flavor for the first scoop? Two, no, not two, 24, right? How many choices for the second scoop? Well, now it depends. It depends on whether I'm allowed to get the same flavor twice or not. And the phrase we're going to use is this, whether repetition is allowed or not. So I'm gonna make a little heading here with repetition, I have 24. If I'm allowed to repeat, if I'm allowed to get chocolate and chocolate, and why wouldn't I ever want to get chocolate and chocolate? Especially if it's the dark chocolate gelato, if you haven't had that, you need to do yourself favor and try that sometime. Or without repetition, only have 23 choices. How do I know which is which? This question is not clear. The questions will be clear. Either they'll say without repetition or the situation that they'll give you, you'll say, oh yeah, I can't pick the same thing twice because that's just stupid. They'll just know. How many different cones are possible? Well, with repetition, without repetition. When I'm during this unit area, I'm gonna give you certain tools, certain tricks of the trade. And one of them is simply drawing blanks. Miguel says, I'm good at drawing a blank. No, no, drawing blank lines. Drawing blank lines, I mean here. What I mean is if they have three choices, I'll draw a blank line for each choice. And then I'll ask how many choices do I have for the first option, three of them? How many choices do I have for the first flavor, 24? How many choices do I have for the second flavor, 24? How many choices do I have grand total, three times 24 times 24, which is what? 1,728. And hopefully you can already see Miguel, we're well beyond wanting to list all those. Like the tree works as an initial step. No, way too, 1,728 branches? No. What about if I'm not allowed to repeat? 1,656, is he right? Yeah. Which is the correct answer? Well, when you order ice cream, is there a rule that says that you can't order the same flavor twice? So I would argue that this is the correct answer, but sometimes the question will clearly suggest that you're not allowed to have a repetition. A computer store sells five different computers, three different monitors, five different printers and two different multimedia packages. How many different computer systems are available? Well, there's four different steps for making choices. I have five choices for the first option, three choices for the monitor, five choices for the printer, two choices for the multimedia package. And I'm pretty sure this is really 15 times 10. See it? I think it's 150. And this gives us the fundamental counting principle. It says this. If one item can be selected in M ways, and for each way a second item can be selected in N ways, then the total, the two items can be selected in M times N ways. The fundamental counting principle says when you're making choices, you multiply. Example, how many different two-digit numbers are there? Well, now I need to think a little bit. How many choices do I have for the first digit? It can start with a one, or a two, or a three, or a four, or a five, or a six, or a seven, or an eight, or a nine. But it can't start with a zero. That's not a two-digit number. So how many choices do I have for the first digit? Nine. How many choices do I have for the last digit? 10, because I think I can end in a zero. How many two-digit numbers are there? We're gonna add a part B here. How many of those are odd? Let's find out. I would say I have two choices. Now, whenever they give me some kind of odd, one, three, five, seven, nine. Yeah, sorry, should have explained that for my ESL student. Even two, four, six, eight, zero. So odd numbers, the restriction to me, what makes an odd number an odd number? How can you glance at a number and tell me right away whether it's odd or even? What do you look at? If you see this, how do you know it's odd? Does that tell you it's odd? No, does that tell you it's odd? What tells you it's odd? Okay, the restriction, or the key, I'll call it the restriction, the condition, it's the last number. I always do the restriction first. So even though I've written it like this, I'm gonna deal with this. I'm gonna say, how many choices do I have for odd numbers there? Five, one, three, five, seven, and nine. I have five choices. How many choices do I have for the first number? Can I have a one there? Can I have a two there? Yes, can I have a three there? Yeah, in fact, you know how many choices I have there? Nine, now somebody said 45, it's half of that, which it is, but I've just confirmed it or proved it, and they're gonna get more difficult than this. So the first tool, the first skill, the first little trick of the trade is drawing blanks. Yes, Miguel, you're good at drawing blanks. The second trick of the trade that I often use is the idea of a grab bag or a little scrabble bag with scrabble tiles, like example three. Example three says, in each case, how many two-digit numbers can be formed using the digits zero, one, three, five, seven, and nine if repetitions are allowed and be if repetitions are not allowed? I honestly very often draw a little grab bag like that and I'm gonna put the number zero, one, three, five, seven, and nine like that, so I can kind of visualize until I get good at this. And then I'm actually going to imagine reaching in and pulling out a number. So how many choices do I have for this first number? One, two, three, four, five, six, I disagree. What can't I pick? Okay, you have to look at this very carefully. What can't I pick, Miguel? Okay, so how many choices do I have for the first number? Five. And then I would say, let's suppose I picked one, not a zero, let's suppose I picked a nine. I would cross that out. And then I would say, now how many choices do I have? Tyler, still five, because there's no other restriction, right? Doesn't say the last number has to be odd or even. So I have five. How many different two-digit numbers can I form if repetitions are allowed? Ah, it's not five. If repetitions are allowed, I'm still allowed to pick that nine, aren't I? So how many choices do I have? How many choices do I have for the second number? Six, there's 30. Compare that if repetitions are not allowed, if I'm not allowed to pick the same number twice. I would start out by saying, how many choices do I have for the first number? Five, how many choices do I have once I've picked, for example, that nine? Five, so here's my question. Many numbers repeat. How many numbers have a repetition? Pat, five. If there's 30 grand total, 25 without repetitions, the remaining five must have repetitions, from page. A true-false test has seven questions. Suppose students answer each question by guessing randomly. How many possible answers are there for each question? On a true-false test, how many answers are there for each question? Two, how many different patterns are possible for the answers? Well, there's one, two, three, four, five, six, seven different questions. Amy, how many choices are there for the first question? What about for the second one? What about for the third one? For the fourth one? Amy, can you find a shorter way to write that instead of writing two times two times two times two times two times two times two? Ah, very nice. Two to the seventh, which is what, 128? Now, if this was an actual test and there's 128 possible ways of making up answer keys, how many of those are the correct answer key? Only one. What's the probability of getting perfect? One out of 128. Come on, you kinda wondered, haven't you? Or better yet, instead of true and false, multiple choice. A multiple choice test has seven questions with four possible answers for each question. Suppose students answer each question by guessing randomly. How many possible answers are there for each question? Four. How many different patterns are possible for the seven questions on the test? It's gonna be four times four times four times four times four times four times four. It's gonna be four to the seventh, which is what? The odds of getting perfect on a multiple choice test by guessing one in 16,384. And of course, the moral of the story is, study. Now, a better question, which we can't answer yet, is Mr. Dewick, what are the odds of passing of getting four or five or six or seven? That's a much more difficult question, tuned in about a month from here. And we will figure that out. The fundamental counting principle, now we just need to look at some weird ones. Can you go to your workbook, please, to page, I believe, 378. Page 378. Some of these get nasty pretty fast. The most common error is you get an answer twice as big as you're supposed to. It means you're double counting without realizing that you're double counting. And I still do that once in a while too. Those of you that purchased the answer keys, of which there's still six left by the way, this is where you'll find, I think, it very helpful because the answer keys, he does show you his steps. In fact, this is where I learned how to do most of it because I had only done about three weeks of this in college and here we're trying to teach it to the grade 12 when this came out in 2001. So determine the number of distinguishable four-letter arrangements that can be formed from the word English if. So I'm gonna visualize my little grab bag here, English. Right now, I'm picking my words very, very carefully. I'm making sure that none of them have the same letter twice because that's gonna be tomorrow's lesson, next class's lesson. What if you got two E's, we'll deal with that later. How many letters are we picking grand total, Justine? We've got four, okay? So if letters can be repeated, one, two, three, four blanks. If letters can be repeated, what that's saying is you pick a letter, look at it, and then you put it back in the bag and shake things up again. I think it's gonna be seven choices, seven choices, seven choices, seven choices. It's gonna be seven to the fourth, which is 49 times 49, 2,500, 2,398. No, what's, what, 2,401? What if no letters are repeated? So we're not allowed to have repetition, which means when I pick a letter, it stays in my hand and there are no further restrictions. So I'm gonna draw one, two, three, four blanks. Irwin, how many choices do I have for the first letter? Seven, now I hold that in my hand. How many choices do I have for the next letter? Six, then, then, seven times six times five times four. Five times four is 20, seven times six is 42, 42 times 20 is gonna be 840. Someone double-check me, 840, yes? Okay. Now they're gonna give me a condition. Dylan, can you read to me part two what's the condition they gave us here? Now, I will always do the condition first, even if it wasn't the first letter, if they said the second letter or the fourth letter, I always deal with that first. I'm still gonna draw one, two, three, four blanks. What's the condition? The first letter has to be what, Dylan? So I'm gonna show that by drawing an E underneath the first blank and I'm gonna look at my grab bag and I'm gonna say, how many choices do I have for a letter E? It's a trick question. How many ways are there to get an E? Only one. So I've picked an E, it's in my hand. Now how many choices do I have left for the next letter? Six, then, then, five, then, four. If the first letter has to begin with an E, it looks like I have a total of 20 times six, 120 choices. Okay. We're gonna come back to the third one in a second, we're gonna do the fourth one. The first and last letters must be vowels for my ESL students and actually for all of you. Vowels are A, E, I, O, and U and never Y, not for the purposes of the provincial exam and not for the purposes of my test. It's gonna be five vowels, A, E, O, I, and U. Those are the vowels. What do we call the non-vowels? Consonants, okay? So one, two, three, four. It says the first and the last letters must be vowels. How many choices do I have for the vowel? What vowels are there here? How many choices do I have? Two. Now the last letter also has to be an E or an I. If I put an E or an I at the front, how many vowels are now left for me to pick from? How many letters am I holding in my hand right now? Two grand total, the E and the I. How many letters are left for this choice? Five, then four, three. The word must contain a G. Does it say it has to be at the front or at the end? It just has to be there somewhere. So this one I'm gonna do, I'm gonna draw one blank separately and then three blanks here and this is gonna be the letter G. How many choices do I have for the letter G? One. How many choices are remaining for the rest of them? So I've got a G in my hand. How many? Six, then five, then four. Except this would be if G was in the front. In fact, I can put the G right there, right there, right there, right there. How many choices do I have for a location for G? 20 times 24. 480? Yeah. These are already considered fairly tough. These are completely fair game. These are fair game as a curve ball, but not 15 of these on your test. We've already done the multiple choice. Let's do example four. So as the telephone number is allocated to subscribers in a rural area, it consists of one of the following. The digits 3, 4, 5, followed by any three further digits or the digit two, followed by one of the digits one to five, followed by any three further digits. Kinda weird, but we'll live with it. Oh, do you? Looking for me? You're just the one for the ride. Okay, we'll be done in about five minutes. Like you to draw a little arrow pointing to the word or and in combinatorics, the word or really is the same as adding a plus sign. In other words, when I read this question and I see the word or in my mind, actually on my paper, I'm doing that already. Says telephone numbers consist of one of the following. The digits 3, 4, 5, followed by any three further digits or plus the digit two, followed by one of the digits one to five, followed by any three further digits. Yeah, let's walk through this. What has to go right here? 3, 4, 5, because it has to go there, what it's really saying is it's only got one choice. You gotta put it there. How many choices do I have here? Well, it says any three further digits and it doesn't say that we're not allowed to repeat. It doesn't say we can't use zeros. So I think we have 10 there, 10 there and 10 there or the digit two has to go there. How many choices do I have for the two? Only one choice, followed by one of the digits one to five. How many choices do I have if I have to put one of the digits from one to five there? Five, followed by any three further digits. I got 10 to choose from, 10 to choose from, 10 to choose from. This one is going to be 1,000 plus 5,000. I think if I've done the math right in my head, it's a 10 times table, that I can do 6,000. Classic question that we love to throw at you is license plate questions. So car plates in an African country consist of a letter other than I or O, followed by three digits, the first of which cannot be zero, followed by any two letters which are not repeated. So it looks like we have six choices to make along the way here. Any letter other than I or O, so I'm gonna write underneath here, I comma O and I'm gonna draw a big X through them, not I or O, like a non-smoking sign. How many choices do I have then? 24, because there's 26 letters in the alphabet. Followed by three digits, the first of which cannot be zero. So this can't be a zero. How many choices do I have? Is there any restriction here? 10, any restriction here? 10, followed by any two letters, but you can't repeat. So how many choices do I have for the first letter? 26, then 25. Did say any three letters, I can bring that I and O back, apparently. How many license plates are possible in this African country? I don't think so. 900 times, ah, read me the digits, one, four, one, four, zero, four, 14 million, 40,000. Okay, yep, louder please. The first letter, like here, it doesn't say that I can't use I or O over here, so I assume it says any letters, any two letters, so I got 26 to pick from again. Yeah, I think it means that I can't use the second over here, I can't repeat this letter. I don't think it's saying that I can't use this one, and if it did mean that, I think they'd phrase it that way, okay? Two more, we're done. It says, consider a six-digit numeral note. If you start with a zero, that's a five-digit number, so no beginning with zero, and I'm not gonna tell you that from now on, I'm gonna say to you, when I ask for a three-digit number, don't start with zero, please. How many odd six-digit numerals have no repeating digits? So we're gonna have one, two, three, four, five, six. And just so I can visualize what's going on, I'm gonna do my little scrabble bag here. Zero, one, two, three, four, five, six, seven, eight, nine. There's my digits. What's the restriction here? Odd, what makes an odd number an odd number? So it has to be a one, or a three, or a five, or a seven, or a nine. How many choices do I have for this last one then? Let's assume, because it says we're not allowed to repeat, so now we're gonna cross off numbers along the way. Let's assume we got a three. How many choices do I have for the first number? No, I don't have nine. Why eight, why not nine? Okay, I can't start with a zero. Can't start with a zero. So I look at what's left, but I can't start with a zero. I have eight choices. Let's suppose, as an example, Alex, that it was a four. How many choices do I have left now? Still eight, because I can have a zero there. Let's suppose it was a nine. How many choices are left now? Seven, then, six, then, five. Eight times eight times seven times six times five times five. Doin' my head? That's a 40, that's a 30, so that's 1200. 1200 times 56, yeah, I can do this in my head. Double 56 is gonna be 112, 112,000? No, really? No, no, no, no, no, Mr. Dewick, you can't be off that much. I doubled it, I got an extra two in there. Like I said, my problem is double counting sometimes. I hope I don't trust you, Miguel, but, you know, 67,200, good gosh, Mr. Dewick, way off. It is, if I had any kind of an ego. Oh, sure, go ahead. Oh, no, I never make mistakes, believe me. How many even six-digit numbers? This question is tougher, and the reason this question is tougher is, what makes an even number? What can we end with? Rattle him off to me, Pat? He said, zero, two, four, six, or eight. What can't I start with? Zero, the fact that zero is in both restrictions is gonna be a problem. I can't start with a zero, and I have to end possibly with a zero. In fact, the only way to do this is to look at two cases, end in zero, or end in two, four, six, four, six, or eight. And remember earlier I said the problem arises when one of our categories, one of our scrabble tiles happens to be in more than one condition. And the best way to do that then is to say, well, let's look at what happens if it's in one, or what happens if it's in the other. And what does the word or mean? And once again, I'm gonna do my scrabble tile here. Zero, one, two, three, four, five, six, seven, eight, nine. We're doing six digits, so one, two, three, four, five, six. One, two, three, four, five, six. And in this first one, we're gonna end in a zero. How many choices do I have to end in a zero? Only one. And the reason that's important is, okay, I just picked a zero, I'll hover my little mouse above it. How many choices are left for the first number now? Not eight, I don't think, why? Why nine this time? Because I've already removed the zero from my scrabble tile. I now have nine choices. Let's suppose I got a two. So I've used the first two. How many choices are left? Eight, then, seven, then, six, then, five. That's if I end in a zero. Or I could end in a two, four, six, or eight. And the reason we're having to separate these again is because zero happens to be in both restrictions, in both conditions. How many choices do I have for a two, four, six, or eight? Four. Let's suppose I picked a two, this guy. How many choices do I have for the first digit? Careful. Eight, because the zero is still in the scrabble tile bag. So let's suppose now I picked a two and I picked a one. How many choices do I have left now? Eight again, then, seven, then, six, then, five. And I get, what was that, 15, 120? Plus eight times eight times seven times six times five times four. 53, seven, 60, and I get 68, 800, and 80. Well beyond our ability to list them and count them. I skipped example six, because I figured example seven kind of covered it. Just really quickly, if they say that you have to be less than 400, how many choices would you have for your first digit here if we're doing three digit numbers and you have to be less than 400? Two, and then whatever and whatever. What's your homework? Yep, the fact that zero is part of the restriction when it says even, zero is included in there, and anytime I'm listing numbers, I can't begin with a zero. Zero appears in two restrictions. You can't start with a zero, make sure you take that into account. Oh, and this time, since it said even numbers, oh, one of your options is you have to be able to end at a zero. There's also a two, four, six, or eight, but I have to look at what if I end at a zero? If I end at a zero, can I possibly start with a zero? Nope, because I've used it up already. That's why I had to list them separately. Look at two, four, six, or eight at the end where you can start with a zero, or end at a zero, which means you can't start with a zero, because you nine choices there. That's the big issue. This is already considered borderline too tough for the provincial exam. In other words, these are, they stay fairly basic for a while, okay? What's your homework? Number one, number two. Take a look at number two. Gonna give you a hint, the answer to number two when it says how many possibilities are there for the score at the end of the first period. The answer is not 18. It's not six times three. I'll let you think about it. Number four, five A, five B, I, five B, three, five B, part four, seven. Three letters followed by three digits. Try eight. Nine A is good, nine B is good, nine C is good, nine, 12 A, 12 B, and we'll pause there, okay? And then next class remind me to show you the ABC menu, restaurant menu that I thought initially was mathematically incorrect but turned out to be right.