 And welcome back. Today we're going to talk about permutations and combinations. Actually, what I'm going to do is I'm going to go back one page. One thing that we have to do before we do permutations and combinations is understand a little bit of vocabulary behind them. Now, factorial, this is something we're going to see in the formulas when we use permutation and combinations. Factorial simply means the product of the natural numbers less than or equal to that number. So, for example, this is five factorial, five factorial. There we go. Okay. And it's just an exclamation point. The factorial symbol is just an exclamation point. So, in English, it means one thing and mathematics will mean something totally different. Five factorial simply just means take five times four times three times two times one. Take this number and all the preceding numbers and multiply them together. And in this case, 120. Now, one special case, though, zero factorial is just going to be equal to one. Okay, that's just a kind of a special no. Factorial kind of breaks down, doesn't really work if we only get to zero. So, we just kind of say, oh, special case, zero factorial is just one, just off to the side there, the last trick. Anyway, permutation. So, the difference between permutation and combination breaks down to this. Permutation is a selection of a group of objects in which order is important. So, this would be something like first, second, or third place in a race or something like that. Order is definitely important because whoever gets first, second, and third, say, for example, if Johnny got first and Sam got second and then Phil got third, that's actually different from if Phil got first, Johnny got second and Sam got third. Those are two, these right here, two different events that could happen. Okay, and so for permutations, it takes into account that these two are going to be two different events. On the other hand, combinations is a group of items in which order does not matter. So, for example, if I want to choose three people, let's actually write out the word, if I want to choose three people to say, go get something for me. So, maybe if I'm, okay, so you're probably watching this, you're probably a student. So, if your teacher asks three people to go to the library, three people to go to the library. Okay, so the thing is, if that person could either pick Johnny, Sam, or Phil, Phil, Johnny, or Sammy, or Sammy, John, and Phil. I mean, there's a bunch of different ways that the teacher could pick these three. But the thing is, is that this is all one group. With combinations, order does not matter. So, the thing is, it doesn't matter what order the teacher picks all of these kids in, it's still just going to be one group. It doesn't matter if he picks, if the teacher picks Johnny first, or Phil first, or Sam first, or something like that, it doesn't really matter. It's just a combination of kids that are going to go to the library. Go get some paper, go get a book, or whatever the case is. Okay, so permutations, order really does matter. Combinations, on the other hand, order does not matter. So that's the big difference between both of them. Okay, now on to a couple of examples. Okay, this is the number of permutations of N items that are taken R at a time. Alright, so here's the thing. This is permutations. This is a permutase example. This is the formula that we use for permutations. Okay, a little bit different notation here. Permutation for the big P, N is the number of items that are taken at a time. You can think of N as your big group. N is your big group of items of people, of kids, of whatever you're talking about. R in this case is how many you're selecting. Okay, let's put selection. So, I'm not spelling that right. There we go. Spelling is not my strong suit. Here we go. Selection. There we go. Alright, so we're going to go ahead and get to the example. How many ways can a student government select a president, vice president, secretary, and treasurer from a group of six people? Okay, now the first thing you have to ask yourself is, is this a permutation or a combination and why? Okay, now obviously this is going to be a permutation, but we'll go over why. This is a permutation because in this case order does matter. So, I'm going to go back to my earlier group. So, Johnny, Sammy, Phil, and Trevor. Okay, those are the four people that are chosen for president, vice president, secretary, and treasurer. But if I have Trevor, Johnny, Phil, and Sam, so Trevor is now the president. Johnny is the vice president. Phil is the secretary. Sam is the treasurer. That's a totally different group. Yes, the same people were chosen, but again that's a totally different group. So, in this case it does matter what order we choose these kids in. So, in this case since order does matter, this is going to be a permutation. Alright, so what I'm going to do is use this notation. Here's use this little formula to figure out how many different groups can be made. Okay, I have a group of six people. If I choose a president, vice president, secretary, and treasurer from those six people, how many different ways can I do this? Okay, so this is what the notation is going to look like. P, and then now N is your big group. My big group of people in this case is six. R is my selection. In this case I'm choosing four people. Okay, four people. I'm choosing a president, vice president, treasurer, secretary, and treasurer. Alright, so that gives me four people. So this is what the math is going to look like. So I'll have six factorial on top, six exclamation point on top, and then I have N minus R, parentheses factorial. So this is going to be six minus four parentheses factorial, which ends up being, which ends up being six factorial over two factorial. Six minus four is going to give me two. Get rid of this example over here. It's going to get in my way. Alright, so now with six factorial and two factorial, I'm going to write this all out. Now, as you get better and better with these permutations and combinations, you won't have to write all this out. You realize here in a moment there's some numbers going to cancel, but that's for when you are more experienced after you go through more examples. So in this case, six times five times four times three times two times one. So that's six factorial. On the bottom, it's going to have two factorial, which is just two times one. Okay, now one thing that you notice is that the twos are going to cancel and the ones are going to cancel. Okay, now this is what I was talking about earlier. Sometimes you're not going to have to write all the numbers because you realize that some of the numbers are actually going to cancel. So in this case, I have six times five times four times three, and then that's it. This is going to tell me how many different choices I have for President, Vice President, Secretary, and Treasurer. So this is going to be 30 times 12. Okay, 30 times 12 is 360 different choices. I have 360 different choices for my President, Vice President, Secretary, Treasurer out of the six people. Okay, so that's an example of a permutations type of problem. Okay, now let's go over a combinations problem. Now let's go over a combinations problem. So the number combinations of N items taken R at a time. So notice that this formula that we have is actually very, very similar to what we had in the last page. Okay, but in this case, we're doing combinations this time. Now N and R are basically the same. N is going to be your big group. Okay, so this is the number of people you're selecting from or this is the total number of geese or whatever the problem asks for. R in this case is how many you're selecting. So again, your selection. Okay, how many geese you're selecting or how many people you're selecting or whatever the scenario is. Now the only big difference here is we do have this R factorial right here on the bottom that we have this R factorial here on the bottom that changes up our combinations just a little bit. Okay. Alright, so here's our example. The swim team has eight swimmers, eight people on the swim team. Two of them will be selected to swim in the first heat. How many ways can they be selected? Okay, so I got eight people on my swim team. I'm going to select two of them to go into the first heat of a race. Now the first thing you got to ask yourself is this is this going to be a permutation or a combination type of problem? Okay, well, this is actually a combination type of problem because if I have if I have Trevor and Sam on my swim team and they're selected. Well, if I also choose Sam and Trevor, this is actually the same. This is actually only one group. Okay, now Trevor and Sam, Sam and Trevor, it doesn't matter who gets chosen first. Both of them are just going to go into the first heat. So yeah, it really doesn't matter who I choose first. All it matters is that I chose both of them. So again, what what is what combinations takes into account is that these two choices is actually one group. So notice that this R factorial on bottom, that little extra that we have there actually takes care of that for us so that we aren't doing we aren't counting multiple selections as as as multiple choices. These two selections are actually just going to be one group. Okay, so here's the math behind it. So for for a combination type of problem and again order does not matter here. I have my big group which has eight so I'm I have eight swimmers on my team and then two of them I'm choosing two at a time to go into my first heat. So this is what the math looks like. So I got eight factorial on top. On the bottom I have two factorial there's my R factorial there inside the parentheses I still have eight minus two and then factorial. So again, this bottom part that this right here is the same as what we had for our previous problem. The two factorial on the other hand is just a little bit different. So on this case, I have eight factorial on top two factorial on bottom and six factorial on bottom. And these are multiplying together. Okay, let me get rid of this group over here is going to get in my way. Alright, so now what I'm going to do now again, as you get more experience with this, you can probably realize okay, there's some numbers that are going to cancel. But you know what I'm going to write them all out so we can see why they cancel. So eight factorial at eight times seven times six times five times four times three times two times one. And then on the bottom I have two times one. And then all that's two factorial and then I also have six factorial was six times five times four times three times two times one. Notice the sixes and fives and fours and threes and twos and ones are all going to cancel. Again, you might see that beforehand and realize that you don't need to write all of that that might save you some time as you're working through these problems. Okay, but on the other hand, this is eight times seven, which is 56. 56 divided by two is only 28. So there's only 28 different choices, 28 different groups of two kids that I can choose to put into this first heat. Okay, choice says get the grammar right. Alright, there we go. Alright, so that is permutations and combinations kind of put all of them in one video. I know it's a little bit longer, but at least I put all of them in one video. Alrighty, thank you for watching this video on permutations and combinations. And we'll see you next time.