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Lecture Comments (12)

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....?

Choices: Combinations & Permutations

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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

Transcription: Choices: Combinations & Permutations

Hi and welcome back to, 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 team0361

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 formulas0552

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 pick0817

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 150968

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 left1001

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 than1294

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 cancellations2337

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 faces2397

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 or2733

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 and3297

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 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