 Hi there, in this video we will discuss using counting techniques to find all the possible outcomes for probability questions. So one of the biggest obstacles is for instance when someone is having four kids, how do you know that there are 16 outcomes because oftentimes you don't want to have to go in and list all the outcomes to figure out how many are there. So that's where counting comes into play. The first counting rule is the fundamental counting rule and it says that if there are m ways of doing one thing and n ways of doing another, then there are m times n ways of doing both. So for instance you go to a restaurant and there's eight different entrees you can order and three different sides you can order that's eight times three or 24 different meals you're able to get right there for you and you can also think about this if you go someplace and they sell wings they got wings that are bone in or boneless they have different types of sauce different types of spice levels and you can find truly how many different ways you can get your wings done or you can order your wings. So Amy has the following colored shirts and pants. She has shirts that are red blue green white and teal and pants that are black blue and brown. So how many outfits where outfit consists of one shirt and one pair of pants can Amy choose from? What is the probability of Amy randomly selecting the red shirt and black pants? So option one is number of outfits that's part one of this question. How many outfits does Amy have? So she had how many shirts to pick from? Five times. How many pants to pick from? Three. So there's 15 outfits total. I mean you could go through and list them all and write red shirt, black pants, red shirt, blue pants, red shirt, brown pants but that's too much work. So that's 15 outfits. So what is the probability of Amy randomly selecting red shirt and black pants? Well that's one of the outfits. That's one of her possible outfits out of the 15 outfits. So the probability is one out of 15. So instead of listing all the outcomes in the sample space we literally learned that there's 15 of them. A thief stole Jenny's debit card. Her pin number can contain any of the numbers 0, 3, 9 and repetition of digits is allowed. What is the probability the thief will guess her pin on the first try? So a pin contains four digits. So that's four numbers we'll be multiplying together here. So digit 1, how many possibilities were there? Well 0 through 9 is 10 different numeric possibilities. Digit 2, repeats are allowed so there's still 10 digits to pick from. Digit 3, 10 digits to pick from. And digit 4 for her pin number, still 10 digits to pick from. So there's a total of 10,000 pin numbers available for Jenny to pick. What's the probability that the thief will guess the pin on the first try? That the thief will guess the pin number on the first try. Well out of 10,000 pin numbers that the thief could possibly guess from, how many are the correct one? Just a single one of them. This one pin number, that's all we have. So the probability is 1 out of 10,000. That's the probability that the thief will guess the pin number on the first try. Makes you wonder, maybe we need to have five digits in our pin, but then that's more numbers to remember. So next is a little note on notation. The factorial symbol is an explanation mark. It denotes the product of decreasing positive whole numbers. So for instance, if you see 4 with an explanation mark that doesn't mean 4, that means 4 factorial. Yeah, no one's ever that excited about numbers, I don't think. But 4 factorial means you start with the number 4 and you multiply by 1 less each time until you get down to 1. 4 times 3 times 2 times 1. So that's going to give you 24. And then by definition, 0 factorial is 1. Now that we know how factorials work, let's do some more counting principle rules. So our next stop is the factorial rule, where all objects are distinct. Distinct means they're not duplicates of each other. So the total number of different ways to arrange in distinct objects is in factorial. So for instance, if you were an elementary school teacher and you had 15 students in your class, the number of ways you could line them up is 15 factorial. Because you have 15 students, they're obviously all different than each other. They're not duplicates. And all the items are used, all 15 students are used. So the factorial rule, when some items are identical to each other, is you still calculate in factorial, however many items you have factorial. But then you divide by how many times does the first item repeat factorial? So number of times item or thing 1 repeats, and then you divide by how many times the second item repeats factorial, and so forth. So you divide by how many times each item repeats factorial. But you still start off with how many items do you have factorial? So if you have 15, if something is 15 factorial, then you divide by repeats factorial. So in my first example, I assume you have a teacher that likes to give pop quizzes. So a pop quiz is given in a history class. Students are asked to arrange the following presidents in chronological order. That's Jefferson, Grant, Madison, Washington, Jackson, and Lincoln. If a student totally guesses, because they were unprepared, was the probability of guessing correctly. So the probability that a student is correct is there's only one correct answer out of how many possible answers. How many ways to arrange those six presidents? Well, they're six presidents and they're all distinct. So it's simply just six factorials, 6 times 5 times 4 times 3 times 2 times 1. We're going to use a scientific calculator to do this calculation for us. So I'm going to use a Desmond scientific calculator. It's an online free calculator. But if you go to function f, u, n, c, and you type in 6, click the explanation mark. It does the calculation for you, telling you 6 factorial is equal to 720. So 6 factorial is equal to 720. So there's only one out of 720 answers that are correct. So 1 over 720 is the probability of guessing the answer correctly. I suppose if you have a teacher that likes to give pop quizzes, make sure you study. How many distinguishable arrangements of letters can be made from the word attract? So we're going to take all seven letters of attract and we're going to rearrange them and we want to figure out how many distinguishable arrangements are there. Well, I don't want you to sit here all day and just literally take the seven letters and move them around and make all these different arrangements. I'm going to use the factorial rule. So I have seven letters, but I do have repeats. I have two a's and I have what else repeats? Three t's. So number of arrangements, number of distinguishable arrangements will equal. Well, I have seven letters, so we start with seven factorial. But because I have repeats, I divide by two factorial, why two factorial? Because you had two a's. And I also divide by three factorial. Why three factorial? Because I had three t's. So we calculate seven factorial. Once again, feel free to use a calculator. Get 5,040 divided by two times six or 5,040 divided by 12. And you'll actually end up getting 420. There's 420 distinguishable arrangements of the letters from attract. But wait, there's more. There's the permutation rule. Suppose you only wanted to take some of some items and you want to arrange them in a specific order. For instance, maybe you have 10 contestants in a competition and you want to pick first prize, second prize, third prize. How many ways are there to do that? That's where the permutation comes in. So first off, all items have to be different. There have to be in different items available. Can't have anything identical to each other. You select some or are of the in items without replacement. And you consider rearrangements of the same items to be different sequences. So if you have ABC, it's different from CBA. So if you have Allen, Betty, and Charles in first, second, and third place, it's a totally different outcome than Charles, Betty, and Allen first, second, and third place. The order is different. Order does matter. So the actual permutation formula is to say NPRN is the number of items. P stands for permutation. R is how many you pick. And the formula is N factorial divided by N minus R factorial. And we'll use technology to aid us with this calculation. A combination is almost like a permutation except there's one major difference. So with the requirements, there's still in different items available. You still select R or some of the in items without replacement. And rearrangements of the same items tend to be that are the same thing. So the combination of ABC is the same as CBA. So maybe you're picking a three-person committee to make decisions about something in an organization. Well, Allen, Betty, and Charles being picked for the committee is the same as Charles, Betty, and Allen being picked for the committee. The order which they were picked does not matter. What matters is that they're serving on the committee. So if the proceeding requirements are satisfied, so order doesn't matter, then the combination formula is NCRN is number of items you have. C is for combination. R is how many items you're choosing. It's N factorial divided by N minus R factorial times R factorial. So it's a little bit more complicated than the permutation rule. And the reason why is because since order doesn't matter, you have a lot of repeat occurrences or repeat outcomes that you have to divide out, so to speak. Kind of like the factorial rule where you had repeats. So that's why we have a little extra in the denominator of our fraction. All right, so when different orderings of the same items are to be counted separately, we have a permutation problem. So permutation order matters. But when orderings are not to be counted, we have a combination problem. Order does not matter for a combination. Let's practice a little bit. Yeah, first, second, and third prizes are to be awarded to three different people. If there are 10 eligible candidates, how many outcomes are possible? So you're giving distinctive placings out. First, second, third. So as a result, when you're talking about placing, specifically first, second, third, order matters. Allen getting first, Betty getting second, and Charles getting third. It's different than if Betty gets first, Allen gets second, and Charles gets third. It's a totally different final placing there. So order matters, which means we have a permutation. How many people do we have? We have 10. How many people are we choosing? Well, first, second, and third prize. So three. So your job to find how many outcomes are possible is to calculate 10p3. 10 is how many items you have. P is for permutation. Three is how many items you're picking, or how many people. In this case, you're picking. So I'll use the Desmos Scientific Calculator. Once again, you want to be in the FUNC or function area, and you want to click the permutation button in PR, and you'll type 10, 3. How many items you have, comma, how many items you will pick? And you'll notice immediately that you get 720. So that's the answer. 720. There's 720 possible outcomes. So it's permutation because order matters when you're talking about first, second, and third place. Let's talk about the Pennsylvania Match Six Lotto. So winning the jackpot requires you to select six different numbers from one to 50. The winning numbers may be drawn in any order. They may be drawn in any order. Find the probability of winning if one ticket is purchased. So in this situation, because the numbers can be drawn in any order. So if one, two, three, four, five, and six are the winning numbers, it doesn't matter what order they're in. As long as you have one, two, three, four, five, and six as the numbers you selected, you're good. So order does not matter, which means this is a combination. So how many numbers can we pick from, well, 50? One to 50 is 50 numbers total. How many numbers are we choosing? Six. So your one and only job is to calculate 50, choose six, 50, see six. It's a combination that says order doesn't matter. The winning numbers may be drawn in any order. So if 50 choose six, we'll use our calculator. We'll choose the combination and see our button. We'll type in 50, comma, six. And look at that, 15,890,700. That's a lot of possibilities. So we have 15,890,700. They don't want to know just how many possible outcomes there are. They want to know the probability of winning if one ticket is purchased. Well, out of 15,890,700 possible tickets that could be created, that's six different numbers being selected from one to 50, only one of them will be the winning number. So that's the probability. One over 15,890,700. If you want, you can divide those two numbers to see truly how small that probability is. Yeah, it's definitely not likely that you're going to win, but it could always happen, right? So that's all I have for now. Thanks for watching.