For more information, please see full course syllabus of Probability

For more information, please see full course syllabus of Probability

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### Choices: Combinations & Permutations

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Choices: With or Without Replacement? 0:12
- Choices: With or Without Replacement?
- Example: With Replacement
- Example: Without Replacement
- Choices: Ordered or Unordered? 4:10
- Choices: Ordered or Unordered?
- Example: Unordered
- Example: Ordered
- Combinations 9:23
- Definition & Equation: Combinations
- Example: Combinations
- Permutations 13:56
- Definition & Equation: Permutations
- Example: Permutations
- Key Formulas 17:19
- Number of Ways to Pick r Things from n Possibilities
- Example I: Five Different Candy Bars 18:31
- Example II: Five Identical Candy Bars 24:53
- Example III: Five Identical Candy Bars 31:56
- Example IV: Five Different Candy Bars 39:21
- Example V: Pizza & Toppings 45:03

### Introduction to Probability Online Course

### Transcription: Choices: Combinations & Permutations

*Hi and welcome back to www.educator.com, these are the probability lectures - my name is Will Murray.*0000

*Today, we are going to talk about making choices and that is going to lead us into combinations and permutations.*0006

*I want to jump right in here.*0012

*There are lots of problems in probability where they say something*0014

*like how many different ways are there to choose from?*0019

*These problems are some of the most confusing ones in probability.*0025

*The reason is that the wording is very subtle and there are two very important distinctions*0030

*that you have to ask about every one of these choosing questions.*0037

*Let me try to walk you through that and in particular,*0042

*I want to draw attention to these two very subtle distinctions.*0046

*Sometimes it is very hard to tell from the wording of the problem but it makes a big difference to the answers.*0051

*Those two subtle distinctions are, are you choosing with the replacement or without replacement.*0058

*Are you making an ordered choice or an unordered choice?*0064

*I want to explain those, explain the differences between those and*0068

*give you some examples of each one so that you can understand what the difference is.*0073

*When you get probability questions, you can make sure that you are understanding the question*0079

*and that you are answering the right question with the right formula.*0084

*I will explain all those differences and then give you all the formulas*0087

*that you need to answer any combination of these questions.*0091

*And then we will work through some examples and hopefully,*0095

*it will all start to be a bit clearer to you by the end of the lecture.*0098

*Our first question here is when you are choosing several things,*0102

*are you choosing with replacement or without replacement?*0108

*The question here is can you choose the same thing more than once?*0116

*If you can choose the same thing more than once, you are choosing with replacement.*0120

*That means after you choose something, that choice goes back into the pool and you can choose it again if you like.*0126

*It gets replaced in the pool and you can choose it again if you like.*0133

*I have couple of examples here to show the distinction between those.*0138

*Suppose you are buying bagels in a bakery and you are choosing do I want an onion bagel?*0144

*Do I want a blueberry bagel? Do I want to a poppy seed bagel?*0148

*And you are going to buy a bag of bagels and take them back to your friends.*0153

*You will say, first, I’m going to buy onion bagel.*0158

*Now, can you choose another onion bagel?*0161

*Sure, because the bakery has a whole shelf of onion bagels.*0163

*That is choosing with the replacement because after you choose the onion bagel,*0166

*you can still choose more onion bagels if you want to.*0171

*Here is an example of choosing without replacement.*0176

*You have a bunch of athletes on the side of a basketball court and*0180

*you want to choose 5 people to be your basketball team.*0185

*You pick the first person and then you want to go back and choose some more,*0190

*can you pick the same person again?*0194

*No, because that person is already on your team.*0195

*That person does not get replaced in the pool after you have chosen that first person.*0200

*That is choosing without replacement.*0208

*We will have different formulas based on whether you are choosing with replacement or without replacement.*0211

*That affects the answer, whether you can make the same choice more than once.*0216

*That is the first distinction.*0221

*Whenever you get a choosing problem, you have to say am I choosing with replacement or without replacement?*0222

*You have to understand that before you can even start to calculate the answer.*0229

*That is the first thing you want to ask, with or without replacement.*0234

*The second thing you want to ask is ordered or unordered?*0237

*Let us talk about that one next.*0241

*And then after that, we will get into some formulas and some actual examples.*0243

*We will calculate some actual problems.*0249

*The second decision you have to make based on the wording of the problem is, is this an ordered or unordered selection?*0252

*In other words, if I choose this thing and then that thing,*0262

*should that be considered different from choosing that thing first and then this thing?*0266

*That is very subtle, it is often not obvious from the wording of the problem.*0270

*But if you are counting those differently then it is an ordered choice.*0275

*If you are counting those to be the same, it is considered an unordered choice.*0280

*Let me give you an example of that.*0286

*It is 2 very similar examples but it will show the subtle distinction.*0287

*Let us suppose we are picking a basketball team and we got 20 people sending on the sidelines*0293

*and we want to pick our basketball team.*0301

*A basketball team is 5 players.*0302

*You have to put 5 people on the court for a basketball team.*0305

*Let us suppose, first of all that this is just a casual, friendly, pickup basketball game in the park.*0310

*We just go out to the park, there are no formal formations.*0316

*We are just going to get 5 people, they are going to run onto the court.*0322

*They are going to throw the basketball back and forth and somebody is going to shoot the ball for the basket.*0325

*That means we are going to pick 5 people and we are going to throw them onto the court, just sort of randomly.*0331

*That is an unordered choice because if I pick Tom, and then Dick, and then Harry to be my basketball team,*0336

*or if I pick Harry and then Tom and then Dick to be my basketball team, it is still the same 3 guys running on the court.*0345

*They are still going to run up and down the court.*0353

*It is the same basketball team either way.*0355

*If I pick 5 guys in one order.*0358

*If I pick 5 guys in a different order, it is still the same basketball team*0361

*because we are not being very formal about this.*0365

*By contrast, suppose we are going to play a formal game, this is a regulation game*0369

*where everybody got positions and everybody is going to stick to their roles.*0374

*You may not be an expert on basketball but there are 5 positions in basketball.*0379

*The positions are there is a center, the guy who stands under the basket.*0384

*There is a power forward, there is a small forward.*0387

*There is a point guard and that should have said shooting guard right there.*0390

*Let me just fix that because that is different from the point guard.*0397

*That says shooting guard.*0400

*There are 5 different positions and you are going to pick 5 players to be your basketball team.*0405

*It matters who plays which position.*0411

*This is different from the informal game on the park.*0413

*You can pick Tom to be your center and then you can pick Harry to be your point guard,*0417

*and then you can fill in the other 3 positions.*0431

*If you pick those same people but in a different order.*0444

*If you pick Harry first, that means Harry.*0449

*You just picked Harry to be your center and then you picked Tom to be your point guard.*0451

*In a formal game, that is a different team because putting Harry in center and*0458

*Tom as point guard is a different team from putting Tom at center and Harry at point guard.*0463

*Even though you got the same guys on the court.*0467

*That is an ordered choice, you are counting those to be different configurations.*0471

*That is what it means to make an ordered choice.*0476

*From now on, whenever you get a probability problem and it has to do with choosing things,*0480

*you got to ask does the order make a difference?*0486

*Does it matter if I pick Tom first and Harry second, and then that means I got one team*0489

*with Tom at center and Harry at point guard, that is an ordered choice.*0496

*If we are just playing an unorganized game in the park, and we are just going to throw Tom and Harry on the court,*0501

*it does not matter who goes on there first, that is an ordered choice.*0509

*And we are going to have a different set of formulas for ordered choices.*0513

*Another kind of common example of this is when you are drawing cards for a poker hand.*0521

*If you are playing poker and you get 5 cards out of the deck, you get your poker hand*0527

*and the question is that an unordered choice or an ordered choice.*0533

*And the answer is that is an unordered because for a poker hand,*0536

*you just get 5 cards in your hand and you can shuffle them around after that, if you like.*0541

*The order that you draw the cards does not matter.*0545

*It is an unordered choice.*0548

*We are going to use those two key decisions to get some formulas*0552

*and see where they lead us and do some examples.*0557

*Let us talk about combinations.*0565

*We are going to learn out combinations and permutations.*0566

*These combinations to count the number of ways to choose a group of unordered objects from N possibilities.*0569

*That means we got N possibilities out there and we are going to choose R,*0578

*make R choices from those N possibilities.*0584

*The important thing here is that we are doing this without replacement.*0589

*Once we choose something, it does not go back into the pool.*0593

*You cannot choose the same thing again.*0596

*And we have a formula for the number of combinations.*0598

*We saw this back in one of the previous lectures here on probability.*0602

*I have showed you where this formula comes from.*0606

*There are two different notations that are very commonly used.*0610

*This is the binomial coefficient notation.*0615

*This is known as a binomial coefficient.*0618

*It is called that because if you expand out the binomial theorem, you get these numbers.*0625

*This is the expression that appears as the coefficients in the binomial theorem.*0631

*This is called the binomial coefficient.*0639

*It is read as N choose R.*0641

*When you read this, you say N choose R.*0643

*That terminology reflects the fact that it comes from choosing R things from N possibilities.*0649

*The other notation for the exact same thing, these are really synonymous.*0656

*It is this capital C with N and R are superscript and subscripts.*0660

*Those really mean the same thing.*0667

*N choose R as a binomial coefficient or C of NR.*0670

*Sometimes people even write it as C of NR, that is not common.*0675

*Depending on what textbook you are using or depending what your teachers preferences might be.*0680

*You might use that notation as well.*0685

*Let me emphasize that this binomial coefficient notation is not the same as fraction notation.*0686

*There is no horizontal bar.*0692

*This is as not N ÷ R, it is definitely not.*0694

*This is a separate notation even though sometimes people mix it up with fractional notation.*0700

*The way you calculate a binomial coefficient is using factorials.*0705

*This is an actual fraction here.*0711

*It is N! ÷ R! × N - R!.*0714

*Like I have showed you in the previous lecture, where that formula comes from.*0720

*You can figure it out yourself but that is how we calculate combinations or binomial coefficients.*0724

*An example of that is unordered selection without replacement is when*0732

*we are selecting 5 players for basketball team from the pool of 20 candidates,*0740

*for an informal pickup game in the park.*0745

*Stress here is that we are going to take 5 people, we cannot repeat the same person.*0750

*We cannot pick the same person more than once.*0757

*We are selecting 5 people, really 5 different people and it is like the same person more than once.*0761

*That is why it is without replacement, you cannot pick the same person more than once.*0768

*And it is unordered because this is an informal game in the park meaning that*0772

*we are just going to throw 5 people out on the basketball court.*0778

*It does not matter what order they run out there, they are just going to run out in one big group.*0781

*This is an unordered selection.*0785

*The number of different ways we can do that is 20 choose 5.*0788

*Expanding that out, it is 20! ÷ 5! × 15!.*0792

*15, I got that by looking at this N - R that is 20 - 5!.*0800

*That would be a very large number which is why I'm not going to calculate it out.*0808

*I’m just leaving it in factorial form.*0813

*If you calculated that out, that is the total number of ways you can pick*0817

*your basketball team for an informal pickup game in the park.*0820

*Let us move on.*0825

*Those are combinations, let us learn about permutations which are very similar*0826

*except we are going to change unordered to ordered.*0832

*Let us see how that is different.*0835

*Permutations looks very similar to combinations but the difference here is that permutations,*0837

*you are counting ordered objects.*0843

*You are making it ordered selection.*0846

*You are still selecting from N possibilities.*0849

*You are still selecting R objects and you are still using without replacement.*0852

*You are still cannot pick the same thing twice.*0857

*But the difference here is that we are talking about ordered objects instead of unordered.*0861

*The notation for that is P of NR.*0868

*Sometimes people use P of NR like that but it is not common.*0871

*The distinction here is that before we had an R! here, and here we do not have and R! anymore.*0878

*There is no more R! in permutations.*0889

*There was an R! for combinations.*0892

*Otherwise, it is the same formula N! ÷ N - R!.*0895

*Let us see a quick example of this.*0901

*I try to make a similar to the previous example so that you can see what the only difference is.*0903

*We are going to select 5 players for a basketball team.*0909

*We have 20 candidates, that is all the same as before.*0912

*The difference here is that we have fixed positions for this game.*0915

*We are going to have a center, we are going to have a power forward, and so on.*0920

*We are going to fill in the other positions, point guard, shooting guard, and so on.*0925

*The difference is that now it matters who plays center and who plays power forward.*0930

*That makes it an ordered selection because if we picked Tom to be the center and then Harry to be power forward,*0938

*that is different from picking Harry to be center and Tom to be power forward.*0948

*We are going to use the P formula, permutations, instead of the C formula combinations to solve this.*0953

*The permutations is 20! ÷ 15!.*0960

*That one actually would simplify fairly nicely.*0966

*It would be 20 × 19 × 18 × 17 × 16 and then I do not have to keep going through 15*0968

*because all 15 down through 1 got divided away by the 15!.*0983

*This would be equal to 20 × 19 × 18 × 17 × 16.*0988

*Another way to see that is, first you pick your center and there is 20 people you can pick for the center.*0994

*And then you pick your power forward and there is 19 people left*1001

*to choose as your power forward, and so on down to your last position.*1005

*Maybe that is the point guard.*1009

*At that point, there is only 16 people left, 16 choices for the point guard.*1014

*That is how you can see that the answer is 20 × 19 × 18 × 17 × 16.*1017

*Remember, the distinction here is that we are ordering the position.*1025

*It matters who plays center and who plays point guard there.*1030

*Let us see some general formulas for this.*1035

*The key formulas when you are choosing R things from N possibilities,*1039

*the questions you have to ask are, is it with replacement or without replacement?*1046

*Is it ordered or unordered?*1052

*Once you get clear on which of those categories you are in, there is a simple formula for the answer.*1055

*Ordered with replacement, it is N ⁺R.*1063

*Ordered without replacement, it is P of NR.*1066

*I will just remind you that that was N!/ N - R!.*1070

*Unordered with replacement, that is the most complicated one.*1079

*It is N + R – 1 choose R.*1082

*Unordered without replacement is N choose R.*1087

*Let us go through and do some examples.*1092

*I try to write several examples that all sound very similar to each other*1093

*but it is designed to test your understanding of these key concepts.*1098

*Are we talking about ordered or unordered?*1103

*Are we talking about with replacement or without replacement?*1105

*Let us go ahead and look at those.*1109

*First example, we got 5 different candy bars and we are going to give them to 20 children.*1112

*We would not give any one child more than one candy bar.*1117

*And then the question is, is this with replacement or without?*1121

*Is it unordered or ordered?*1124

*And how many ways can we distribute the candy.*1125

*What we are doing here is we got 5 candy bars, for each candy bar*1129

*we are going to choose a child to give the candy bar to.*1134

*We are making 5 choices here.*1139

*We are choosing R = 5 children because for each candy bar, we choose a child to give it to.*1147

*Each time we choose a child, there are 20 possible children we can give it to.*1160

*That is N = 20 possible children.*1164

*The key points here are, first of all, we do not want to give any child more than one candy bar.*1180

*That means once I have given a candy bar to a child, that child has to leave the room.*1187

*I do not get to pick him again to give another candy bar to.*1193

*That means I'm choosing without replacement.*1197

*Once that child gets a candy bar, that child is not available.*1205

*It is not replaced as a choice for the next candy bar.*1209

*Without replacement, that is the answer to our first question here.*1212

*Second question is, is it ordered or unordered?*1222

*If I choose Susan and then for the first candy bar and then Tom for the second,*1225

*is that different from choosing Tom for the first candy bar and then Susan for the second?*1232

*The clue to the distinction there is in the first sentence, we are giving 5 different candy bars out.*1237

*Because we are giving 5 different candy bars out, let us say I’m giving out snickers and the 3 musketeers.*1248

*I will give out a snickers and then the 3 musketeers.*1254

*If I give the snickers to Susan and the 3 musketeers to Tom,*1258

*that is different from giving Tom the snickers and Susan the 3 musketeers.*1264

*These are different.*1272

*They are different because they are different candy bars.*1273

*These 2 distributions are different, that means that the order matters.*1277

*If I'm going to give out the snickers first and then the 3 musketeers,*1286

*it matters which order I pick the children in.*1292

*If I pick Susan first and then Tom, they are going to get a different distribution than*1294

*if I pick Tom first and then Susan.*1300

*The fact that they are different candy bars means that order matters.*1304

*This is an ordered selection.*1309

*This is ordered without replacement.*1318

*Now, I just look up on my formula table and I see ordered without replacement.*1320

*If you check a couple slides back, you will see the ordered without replacement means*1325

*we use P of NR which in this case is P of 25 because N is 20, R is 5.*1330

*This is 20!/ 20 - 5!.*1343

*Using the formula for permutations, N!/ N - r !.*1351

*This is 20!/ 15!.*1357

*And if you want to simplify that, you can cancel out all the factors of 1 through 15 from the 20!.*1363

*You will get 20 × 19 × 18 × 17 × 16 ways to distribute this candy.*1370

*The other way to think about that is, I got 5 different candy bars.*1381

*I look at the first candy bar and I say there is 20 children.*1386

*I can give that first bar to then I would give that first bar away and*1392

*I send that child out of the room because I do not want that child to get another candy bar.*1397

*Give the second bar out, I got of 19 choices for the second bar.*1403

*I will send that child out of the room.*1407

*I got 18 choices left and I go through and distribute all my candy.*1408

*By the time I give out the last bar, there is 16 children left trying to get that last candy bar.*1414

*Just to focus here on what is important, what we are trying to decide is with replacement or without.*1420

*The way we know it is without replacement is that we do not want to give the child more than one candy bar.*1428

*Once I choose a child, that child does not go back in the pool and he is not replaced in the pool.*1437

*It is without replacement.*1443

*It is an ordered selection because the candy bars are all different.*1445

*It matters who gets which candy bar.*1448

*I look at my chart, I see that without replacement and ordered gives me permutations of N and R.*1454

*I fill in the formula for permutations, simplify it down, and that is the number of ways that I can do it.*1461

*The next example is going to look very similar to this one.*1469

*You want to read it very carefully and see if you notice the difference between the next example and this one.*1472

*It is going to look almost the same, there is one small difference and*1478

*that is going to dramatically change the answer that we get.*1481

*That is really how probability questions go.*1484

*Very small differences in the wording made a big difference in the answer.*1486

*Let us check out the next example.*1490

*We have 5 identical candy bars to distribute to 20 children.*1494

*We do not want to give any child more than one candy bar.*1498

*Is the selection with replacement or without? Is it ordered or unordered?*1501

*How many ways can we distribute the candy?*1504

*This looks almost the same as the previous example.*1508

*Again, we are choosing R = 5 children.*1512

*We are making 5 choices from each time we choose a child, there are 20 children hoping to get that candy bar.*1521

*We have N = 20 possibilities.*1530

*Is this selection with replacement or without?*1541

*Remember, we said we do not want to give any child more than one candy bar.*1545

*That means, once we select the child, give that child a candy bar, he has to leave the room.*1549

*He does not go back in the pool of recipients.*1555

*He is not replaced in the pool.*1559

*Because we do not want to give any child more than one candy bar, this is without replacement.*1564

*No child can have more than one candy bar.*1572

*That is what that tells us right there.*1579

*Is this selection unordered or ordered?*1582

*Remember, last time we have 5 different candy bars.*1588

*This time, we have 5 identical candy bars.*1593

*That means we are giving out 5 snickers.*1597

*I will give you a snickers, I will give you a snickers.*1600

*What that means is, if I choose Susan to get the first snickers*1606

*and then I choose Tom to get the second snickers, they both walk away with a snickers.*1612

*If I choose Tom to get the first snickers and then I choose Susan to get the second snickers,*1618

*they both walk away with the snickers.*1626

*It is the same to them, it is the same distribution of candy.*1627

*As if I had chosen Susan first and the Tom second or if I had chosen Tom first*1631

*and Susan second because the candy bars look the same.*1637

*If they were different candy bars, the order would matter but because the candy bars are identical,*1641

*this is an unordered selection.*1648

*That really depends on the wording of the problem.*1656

*This problem is almost identical to the previous one.*1660

*The only difference is that the candy bars are identical, instead of being different.*1662

*But because of that subtle difference, we use a different formula.*1668

*The formulas, I lay them all on a chart in one of the earlier slides.*1672

*You can go back and look it up.*1676

*The answer for unordered with replacement is, we use combinations to calculate this.*1678

*The binomial coefficient C of 25 or you can also think of this as 20 choose 5.*1688

*That is different notation for the same thing.*1696

*This is from N = 20 and R = 5, the number of choices we are making.*1700

*Remember, the way you expand a binomial coefficient or the combination formula is N!/ 5!.*1708

*Let me write that as R.*1721

*× N - R!.*1724

*I will plug in what the numbers are.*1727

*I get 20!/ R is 5! And 20 -5 is 15!.*1729

*And I think what I will do is I will cancel out some of the terms from 20! With the 15!.*1739

*That will leave me with 20 × 19 × 18 × 17 × 16.*1746

*I’m canceling out all the numbers after that with the 15!.*1753

*I will divide that by 5! because that is still in there.*1758

*5 × 4 × 3 × 2 × 1.*1762

*It looks like there is a lot more cancellation I could do there.*1766

*I think I will go ahead and cancel the 20 with the 5 × 4.*1771

*I will cancel the 18 with 3 and turn it into a 6.*1776

*And then I can turn that 6 cancel that with the 2 and give me 3.*1781

*I get 19 × 3 × 17 × 16.*1787

*It is still a pretty big number.*1792

*I think I do not want to multiply that out.*1793

*But that is the number of ways that I can distribute my identical candy bars to these 20 deserving children.*1796

*Let me emphasize here that this one is almost identical to the previous problem except*1805

*that the candy bars are identical now.*1811

*We are still making 5 choices from 20 possibilities.*1813

*We are still choosing without replacement.*1817

*That is because we do not want to give a child more than one candy bar.*1820

*It means that after we give a child a candy bar, that child is not replaced in the pool.*1825

*That child has to leave the room, cannot get another candy bar.*1830

*The difference is that this is an unordered selection because the candy bars all look the same.*1836

*The children do not really care who gets picked first because they are all going to get the same candy bar in the end.*1843

*The candy bars all look the same.*1850

*Because it is unordered, because it is without replacement, we use our combination formula.*1854

*This is coming from the chart that we had on the slide a few minutes ago.*1859

*I use the combination formula.*1865

*These are two different notations for the same formula but they both expand*1867

*into the same fraction of N!/ R! × N - R!.*1872

*Plugging in N and R and then expanding and cancelling what I can,*1880

*gives me the total number of ways I can distribute the candy.*1885

*If you check back and compare this with example 1, it is almost the same*1889

*with just one word that I changed but the answer is quite different at the end.*1893

*There are fewer ways to distribute this candy than there was in example 1.*1898

*We are going to keep going with this in the other examples.*1904

*I’m going to make small changes but it is going to keep changing the answers.*1906

*The idea is to practice the distinction between with or without replacement and ordered vs. unordered.*1910

*Let us give out some more candy in example 3.*1917

*We 5 identical candy bars to distribute to 20 children.*1921

*We are willing to give some children more than one candy bar.*1926

*If I happen to call the same child twice, that child gets 2 candy bars and that is okay.*1929

*We are being asked is the selection with replacement or without?*1935

*Is it ordered or unordered?*1939

*How many ways can we distribute the candy?*1940

*Just as before, just as with the first two examples.*1945

*This is very similar to those.*1949

*We have R = 5 children.*1950

*We are making 5 choices because I have my 5 candy bars.*1953

*Each time I’m going to pick a child and give that child a candy bar from N = 20 possibilities.*1959

*Because each time I pick a child, there is 20 children to choose from.*1968

*Is this selection with replacement or without?*1979

*That means if I choose a child, can I choose again and give that same child a second candy bar?*1983

*According to the stand with the problem, yes, they are willing to give some children more than one candy bar.*1990

*Based on that, after I choose a child, that child gets to stay in the room*2001

*and is still eligible to get a second candy bar.*2006

*We are choosing here with replacement.*2010

*That child gets replaced into the pool because that child might get to stick around and get a second candy bar.*2014

*Now, our second question is, is unordered or ordered?*2025

*That means if I pick Susan and then I pick Tom, is that different from picking Tom and then picking Susan?*2029

*The key is to look at the candy bars.*2036

*In this case, all the candy bars are identical.*2040

*If I give a candy bar to Susan and then a candy bar to Tom,*2045

*it is going to be the same as if I give a candy bar to Tom and a candy bar to Susan.*2048

*That means the order does not matter.*2054

*Either way they both get a candy bar.*2056

*It is unordered, this is an unordered selection with replacement.*2059

*We can figure out how many different ways there are to do this,*2070

*by looking at our chart that I had a few slides back.*2074

*Unordered with replacement, the formula for that is the binomial coefficient N + R -1 choose R.*2079

*The notation for the same thing is C of N + R – 1 R.*2091

*In this case, N + R -1.*2103

*N is 20, R is 5.*2107

*20 + 5 - 1 that is 24 and R is still 5.*2109

*If you remember our formulas for the binomial coefficients, for combinations, it is 24! ÷ 5! and 24 - (5!).*2119

*That is 24!/ 5! × 19!.*2137

*And if you want to expand this out then on you can solve the 24! or at least a lot of the terms,*2145

*a lot of the factors with the 19!.*2154

*You will just be left with 24 × 23 × 22 × 21 × 20.*2157

*And then from there on, it is 19!.*2164

*It would cancel out with the 19! in the denominator.*2167

*In the bottom still is the 5! 5 × 4 × 3 × 2 × 1.*2171

*I guess we can keep going with that.*2178

*We can cancel the 5 and the 4 with the 20 and then the 3 and the 2, that is 6.*2181

*We can cancel out with 24 and get 6.*2189

*We get 6 × 23 × 22 × 21 ways that we can distribute this candy.*2192

*And then you can multiply these numbers together.*2205

*I do not think it would be that illuminating to multiply the numbers together,*2206

*that is why I’m leaving that one in factored form.*2210

*Let me go back and just make sure it is clear how we got each step of that.*2213

*There are two key phrases that tell you how to calculate this and*2219

*they come from the wording of the problem.*2223

*Probability is so subtle, you really get to read these problems very carefully.*2224

*If there is anything unclear, it is often going to ask your teacher just*2229

*to make sure you know what you are being asked.*2233

*Because the subtleties in the wording really affect the answer.*2235

*Here, we are making 5 choices because I have got 5 candy bars.*2242

*Each time I have a candy bar, I’m going to choose 1 of 20 children to hand it out to.*2247

*There is 20 possibilities for each one of those choices.*2253

*With the replacement, that comes from the fact that I'm willing to give some children more than one candy bar.*2257

*If I give all 5 candy bars to Susan, that Susan’s lucky day.*2264

*We are willing to consider that possibility.*2270

*We want to count that possibility.*2272

*That is why we say it is with replacement.*2275

*After I gave Susan the first candy bar, she gets to go back into the pool.*2276

*She gets replaced in the pool and she gets to hope that maybe she can get a second candy bar too.*2280

*It is unordered because these candy bars are identical.*2285

*The order does not make a difference.*2290

*If I gave Susan a candy bar and then Tom a candy bar,*2292

*that is the same as if I give Tom a candy bar and Susan a candy bar.*2295

*They candy bars look the same.*2300

*They both walk out of the room with the same kind of candy bar.*2302

*Because it is unordered and with replacement, if you look back in that chart from a few slides ago,*2306

*we use this combination formula N + R -1 choose R.*2312

*This is just a different notation for the same thing.*2317

*N is 20 and R is 5, we get this and then this is my formula for combinations.*2320

*Expand that out into factorials and then the 24! has a lot of factors in common with the 19!.*2329

*That is why I cancel those off at this step and then I did some more cancellations*2337

*to simplify the numbers a bit and got my final answer there.*2341

*The 4th example is again going to look very similar to this one.*2347

*I just made a very small change in the wording and you will see that*2350

*it does change the answer in the original wording looks very similar.*2354

*Let us go ahead and see how that one plays out.*2359

*Example 4, we have 5 different candy bars to distribute to 20 children.*2362

*We are willing to give some children more than one candy bar.*2368

*Is this with replacement or without?*2372

*Is it unordered or ordered?*2373

*How many ways can we distribute the candy?*2375

*Again, we are making R = 5 choices because for each candy bar, I make a choice of which child to give it to.*2379

*There is 20 children and my N is 20 possibilities.*2391

*20 children that each time I have a candy bar, I look at 20 possible hungry faces*2397

*and decide which one I want to give the candy bar to.*2403

*I have to say is this with replacement or without?*2411

*The key phrase here is we are willing to give some children more than one candy bar.*2417

*If I give the first candy bar to Susan, I’m willing to give another one for the second candy bar.*2422

*She gets replaced in the pool.*2427

*She gets to go back into the pool and hopefully get another candy bar.*2429

*That means that we are working again with replacement.*2435

*The next question is, is this an unordered or ordered selection?*2453

*I will put just a single star there because that was the first question.*2462

*The second question is, are we working unordered or ordered?*2467

*The key to answering that is the fact that we have different candy bars.*2472

*This time, we maybe have snickers and the 3 musketeers and maybe several other candy bars,*2478

*but they are all different.*2485

*If we pick Susan first and she gets the snickers, and then Tom gets the 3 musketeers,*2488

*that is different from picking Tom first to get the snickers and Susan to get the 3 musketeers.*2495

*That would make a difference, maybe Susan particularly likes snickers.*2501

*In the first arrangement, she would be very happy.*2504

*In the second arrangement, she would be upset because she did not get the candy bar she liked.*2507

*The order really matters, those are two different arrangements there.*2511

*We would be picking 3 more children there or possibly the same children again.*2518

*The order really matters there.*2522

*This is an ordered selection.*2524

*This is with replacement and it is ordered.*2532

*If you go back and look at the chart, how many ways are there to make 5 choices*2537

*from 20 possibilities and it is ordered and with replacement?*2541

*The answer from the chart, this chart was on 3 or 4 slides ago.*2545

*Just scroll back through the video and you will find that chart.*2552

*The chart tells us that the answer is N ⁺R.*2555

*In this case, our N is 20 and our R is 5.*2560

*There are 20⁵ ways of distributing the candy, that is our answer.*2566

*The keywords that you want to look for in the wording are the fact that we are willing*2573

*to give some children more than one candy bar.*2579

*That is how we know it is with replacement.*2581

*The fact that they are different candy bars.*2584

*What really matters, which kid comes first and get which candy bar.*2586

*That is why it is an ordered selection.*2591

*We count Susan and then Tom different from Tom and then Susan.*2592

*Ordered with replacement from our chart is N ⁺R ways.*2598

*That is where we get the 20⁵.*2602

*This one by the way is one word pretty easy to confirm the answer.*2604

*If you think about it, when you got that first candy bar, that snickers bar,*2608

*you look around and you see 20 hungry faces.*2612

*You did make a decision about which child you are going to give it to.*2615

*You have 20 possible choices for that first candy bar then you see the second hungry child*2620

*or the second candy bar, and you look around at that sea of faces again,*2626

*there are still 20 kids clambering for that second candy bar because the first child got to go back into the group.*2630

*There are 20 possibilities to give away that second candy bar and then 20 possibilities for the third bar as well.*2638

*First bar, second bar, and so on.*2648

*There are 20 possibilities for each candy bar and every child gets counted for every candy bar*2652

*because even if you pick a child for the first candy bar, that child can still line up again and ask for a second candy bar.*2660

*You are multiplying together 25 times, we get 20⁵.*2667

*That one was fairly easy to see the answer intuitively.*2672

*It is also possible just to read the answer off our chart and that is what we did.*2677

*We got one more example here.*2684

*It is going to be different from this one.*2685

*We are not going to give any more candy to any more children.*2687

*I try to make it a little different and a little less obvious.*2690

*But again, it is going to be the same key decisions.*2693

*Is it an ordered selection or unordered?*2696

*Is it with replacement or without replacement?*2699

*Let us check that one out.*2701

*This one is still food related.*2704

*I must have been hungry when I was writing this lecture.*2706

*What we are going to do is we are going to go to a restaurant and we got a big party going on at home.*2711

*We are going to buy 10 pizzas from a restaurant, bring them all back home to our hungry guests in our party.*2715

*It is a fairly simple restaurant, they only carry 3 kinds of pizzas.*2721

*They carry cheese pizzas and pepperoni pizzas and the vegan pizzas.*2727

*We want to buy 10 pizzas total and we could buy all 10 cheese or*2733

*we could buy 5 cheese and 2 pepperonis and 3 vegan pizzas.*2739

*We want to figure out how many different orders can we make.*2745

*And in particular, is this selection with replacement or without replacement?*2748

*Is it ordered or unordered?*2753

*Does it matter the order in which we picked the pizzas?*2758

*Finally, how many possible different ways are there to make our order from the restaurant?*2761

*Let us think about that.*2767

*First of all, we are choosing 10 pizzas here.*2769

*We get to make 10 choices.*2778

*I want the first choice.*2779

*I want the first pizza to be pepperoni.*2780

*I want the second one to be vegan.*2784

*I want the third one to be another vegan pizza.*2785

*Then I want to choose one so we get to make 10 choices here.*2788

*We choose 10 pizzas.*2798

*That R is the number of choices that we get to make.*2799

*R is 10 pizzas.*2803

*Each one of those choices, we look around and we see there is only 3 possible pizzas.*2808

*Each one can be cheese, pepperoni, or vegan, from 3 possibilities.*2817

*That is the N there, that is the number of possibilities for each choice.*2827

*N = 3 possibilities.*2829

*That kind of sets up our problem here but the two key questions we have to ask are, with replacement or without?*2837

*Unordered or ordered?*2842

* Let us think about replacement.*2845

*After I choose, I say I want the first pizza to be cheese.*2847

*Does that mean I can still pick another cheese pizza or is cheese off the table now?*2852

*The answer is that I can still pick another cheese pizza.*2856

*Because the restaurant can make as many cheeses I like.*2861

*It can make as many pepperoni, as many vegan as I like.*2863

*It is possible to have more than one cheese pizza in my order.*2868

*That means it must be with replacement.*2872

*How can I write this down?*2880

*We can say, we can get more than one, for example cheese pizza.*2881

*You are making a choice with replacement.*2895

*It is not like after we choose a cheese pizza, they run out of cheese and we cannot buy another one.*2897

*Cheese is still an option, even after we say I want the first pizza to be cheese.*2904

*That is okay, the second pizza can still be cheese,*2912

*if we happen to be great fan of cheese.*2915

*Unordered or ordered?*2918

*Suppose we choose 5 cheese pizzas and then 5 vegan pizzas,*2920

*and we take them all home to our guests at the party.*2927

*First, we can choose 5 vegan pizzas and then 5 cheese pizzas,*2931

*then we take them all home to our guests at the party.*2934

*Will that make any difference to our guests?*2937

*No, they will be happy either way.*2939

*Whatever order we choose these pizzas in, we are going to pile them all together,*2943

*take it back to the party and our guests are going to choose whatever they like no matter what.*2947

*We come home, we returned home with 10 pizzas and it does not really matter*2954

*what order they are stacked up in.*2966

*With 10 pizzas, it does not really matter what order they are stacked up in.*2970

*Our guests still get to pick whichever pieces they like in no particular order.*2979

*This is an unordered selection because it really does not matter when we are making our order at the restaurant,*2992

*it really does not matter whether we pick 5 cheese first and then 5 vegan,*3002

*or 5 vegan pizzas first and then 5 cheese pizzas.*3008

*We are still going to come home with 5 cheese and 5 vegan either way.*3012

*It does not matter to our guests.*3017

*It does not change what our order is at the restaurant.*3018

*What we have here is a selection that is unordered with replacement.*3024

*If you go back and look at our chart which we had a couple slides ago,*3029

*we are going to use combinations to solve that, the way you solve an unordered*3034

*with replacement selection is you use N + R -1 choose R.*3038

*The other notation for that is C of N + R -1 R.*3045

*In this case, the R is 10 and the N is 3.*3053

*10 + 3 - 1 that is 13 -1 is 12.*3057

*The R was 10.*3063

*Remember, the formula for combinations.*3069

*That is 12! ÷ 10! × 12 -2!.*3074

*This one we are going to get a lot of cancellation.*3084

*12! ÷ 10! × 2!.*3086

*If we expand out 12!, we get 12 × 11 and then × 10 × 9 × 8, and so on.*3092

*That one cancels with the 12th or the 10! in the denominator.*3100

*The rest of it cancels with the 10! And we just have a 2 × 1 from the 2! in the denominator.*3105

*The 12 will cancel out the 2 and give a 6.*3114

*66, 6 × 11 is 66 possible orders.*3117

*This number of possible orders at our pizza restaurant actually is not big.*3125

*There are 66 different ways we can distribute our orders between cheese pizzas, pepperoni pizzas, and vegan pizzas.*3132

*Let us recap how we arrived at that conclusion.*3143

*We are buying pizzas at a pizza restaurant and we are buying 10 pizzas which is where we are making 10 choices.*3146

*10 different times we get to say I want a vegan pizza, I want pepperoni pizza.*3155

*Each time we make one of those choices, there are 3 possibilities.*3160

*There is vegan, pepperoni, and cheese.*3163

*There are 3 possibilities, that is where we get R and N there.*3166

*Is it with replacement or without?*3170

*Once you pick a cheese pizza, you can pick another cheese pizza if you like.*3172

*It is with replacement.*3177

*Is it ordered or unordered?*3179

*It do not really matter whether you order the vegan pizza first and the pepperoni pizza second, or vice versa.*3181

*You are still going to come home with a vegan and a pepperoni pizza.*3187

*It is unordered.*3191

*You just look at the chart that we have a few slides back in the introductory slides.*3193

*I gave you a little chart with replacement, without replacement, unordered, ordered, and then formulas for each one.*3198

*The formula for with replacement and unordered was N + R -1 choose R.*3205

*When you plug in R = 10 and N = 3, you will get 12 choose 10.*3213

*That is 12!/ 10! ×,*3218

*There is a small mistake there.*3224

*This is what happened to recap because that 2 should have been 10.*3225

*I did not write on the next step or 12 -10 is 2!.*3230

*12 - 10 that is why I got a 2! on the next step.*3241

*The calculations are still right because it is this step, it was just a little mistake right there.*3247

*12!/ 10! × 2!.*3253

*When you write those out, all the factors of 12! Or most of them cancel with the factors of 10!.*3257

*That is why I did not write the 10! Factors here because they all got canceled with the 10! part of 12! up here.*3264

*Left me with just 2 × 1 in the denominator from the 2!.*3275

*12 × 11 in the numerator cancel the 12 and the 2 and get 6 × 11*3279

*and you get 66 possible ways to make our pizza order at this pizza restaurant.*3285

*That is the end of this lesson on making choices.*3293

*Let me emphasize that this is all about reading these problems very carefully and*3297

*deciding is this an ordered choice or an unordered choice?*3302

*Is this a choice with replacement or without replacement.*3306

*We will do the same thing when you study your own probability problems that have*3310

*to do with making choices and counting things.*3313

*You will ask the same questions each time, ordered or unordered?*3316

*Replacement or without replacement?*3320

*And then once you have answered those questions, we have this nice chart of all the different formulas.*3323

*You just drop it into a chart, you get a formula and then you can calculate the number of ways to make your choices.*3328

*That is the end of this lecture on making choices.*3336

*This is part of a larger series of lectures on probabilities.*3338

*I hope you will stick around and sign up for the other lectures here on www.educator.com on probability.*3342

*We got all kinds of good stuff divided into all kinds of categories.*3350

*We are going to help you get through your probability course.*3356

*Thank you very much for joining me.*3359

*My name is Will Murray and this is www.educator.com.*3360

1 answer

Last reply by: Dr. William Murray

Tue Oct 4, 2016 2:08 PM

Post by Thuy Nguyen on October 2, 2016

Hello Dr. Murray,

In the pizza example, there are 66 possible combinations of choosing 10 pizzas from 3 styles. If I want to know the probability of having at least 2 cheese pizzas from the 66 combinations, then:

r = 8 pizzas that I need to choose, since I already have 2 cheese pizzas.

n = 3 styles to choose from

With replacement and unordered.

3+8-1 choose 8 = 45

Thus P(at least 2 cheese pizzas) = 45/66

Right?

3 answers

Last reply by: Dr. William Murray

Fri Apr 10, 2015 12:39 PM

Post by Anna Ha on April 8, 2015

Hi Dr. Murray,

How would you do this question?

A box contains seven snooker balls, three of which are red, two black, one white and one green. In how many ways can three balls be chosen?

I tried using combinations but it didn't give me the correct answer...

Thank you!

1 answer

Last reply by: Dr. William Murray

Mon Nov 24, 2014 9:50 PM

Post by Jim McMahon on November 23, 2014

Will -- having the trouble advancing the lecture video again. Something about the laptop that I am using I think. Do you have any idea what settings (perhaps Adobe) that might be key to enabling me to fast forward in a lecture? Right now, I have to use a different computer (desktop) to be able to advance to a point further in the lesson. Have played well into the video so it does not appear to be a case of letting the buffer properly load. Any hints would be appreciated.

1 answer

Last reply by: Dr. William Murray

Mon Sep 15, 2014 6:24 PM

Post by Jethro Buber on September 14, 2014

statement: in formula you have 12 minus 2 but should be 12 minus 10. you still nailed it making it equal to 2!.

1 answer

Last reply by: Dr. William Murray

Sat Jul 5, 2014 5:53 PM

Post by Thuy Nguyen on June 28, 2014

How can we choose 10 from 3? That wouldn't make sense. Instead, wouldn't n = 10 pizzas and r = 3 choices? So it would be 12 C 3 for the pizza question, making the answer 220 choices in all? 66 seems too small.

1 answer

Last reply by: Dr. William Murray

Thu Mar 27, 2014 6:31 PM

Post by Heather Magnuson on March 25, 2014

I think that, on example III it should be a 4 instead of a 6 in the answer....?