All language subtitles for 011 A Practical Example of Combinatorics_en

Instructor: Welcome back.

This video is going to be a practical example

where we showcase and apply

most of the knowledge gained in this section of the course.

More precisely,

we are going to use our newly acquired understanding

of combinatorics

to aid us in our quest of ordering something

nobody has ordered before.

So you and your friend, let's call her Amy,

wanna get some Domino's Pizza for your night in.

You open the Domino's app on your phone

and see that the store which delivers to your home

offers several deals.

A One Person Combo Pack,

a Family Deal,

and Double Delight.

The first one consists of a medium pizza,

a small drink, and a dessert for 15.99.

The Family Deal includes two medium-sized pizzas,

a small drink, a large drink, and two desserts for 25.99.

Lastly, Double Delight also includes two pizzas,

two drinks, and two desserts,

and costs 27.99.

The only difference being the two drinks are both large.

Before picking a specific deal,

you pull up the menu to explore the options.

You see they offer a variety of 26 different pizzas,

4 distinct drinks,

as well as 7 types of desserts.

Initially, Amy says she isn't that hungry.

So you look at the One Person Combo Pack.

Now, to construct such a menu,

you need to combine three elements,

a pizza,

a small drink,

and a dessert.

Since all of these have distinct sample spaces,

we consider every part of the menu as a separate position.

Domino's offers 26 pizzas,

so there are 26 different ways

we can choose our single pizza.

Similarly, we have four ways

of picking our preferred beverage

and seven ways to pick our dessert.

This scenario is identical

to the diner example we explored a few lectures ago

where we had to construct a lunch menu

out of a sandwich, a side, and a drink.

Therefore, we know how to find

the total number of different combinations.

We simply multiply the size of the sample space

for each position.

This would result in 26 times 4 times 7,

or a total of 728 possible ways

to fill out this deal.

Now, we use multiplication here

because regardless of which of the 26 pizzas we get,

we can accompany it with any of the four beverages.

Following the same logic,

for any of those pizza and drink combos,

we have seven different dessert options.

Fascinated by the huge variety,

your friend reconsiders and decides to get some food, too,

not because she's hungry,

because she wants to play the combinations game

and increase that number of options you have.

Thus, you're compelled to get one of the two larger deals,

the Family Deal or the Double Delight.

Amy states that since you're getting two pizzas now,

you can get different ones and exchange slices.

Furthermore, she proclaims that

"Since we're ordering two of each

"of the three parts of the meal,

"we now have 8 times as many options."

Having recently mastered combinatorics,

you decide to test if her statement holds true.

To do so,

you must compute the number of different menus you can order

for each of the other two deals.

You examine the Family Deal they offer first.

Recall that it includes two pizzas,

a small drink,

a large drink,

and two desserts

for the price of 25.99.

Once again, you decided to get different pizzas

to get the most out of your deal.

Since the drinks vary in size and desserts are small,

you decide to make no such commitments

about the other two ingredients.

All right.

Let's first look at the pizzas you can choose.

Since you are splitting the two among one another,

no order is involved in this decision.

Thus, you would need to use combinations without repetition

to determine the correct amount of pizza options

this deal provides.

You apply the formulas you learned already,

which results in 26 factorial over 24 factorial

times 2 factorial.

This simplifies to 25 times 26 over 1 times 2,

or 325.

Your conclusion is that there are 325 different ways

of picking the pizzas.

Whoa, that is a much greater variety

than the 26 we had available earlier.

Now, for the drinks.

You can both order the same one.

So repetition is allowed.

Furthermore, since it matters which drink is larger,

this situation requires using variations.

Therefore, we apply the formula

for variations with repetition

for picking two out of the four available options.

By plugging values into the formula,

we see that we have 4 to the power of 2,

or 16 options for the drinks.

Alternatively, you can think of these

in terms of combinations of two events

with separate sample spaces.

By doing so, we get two events,

both of which have sample spaces of size 4.

As we can expect,

the answer would remain the same

since 4 times 4 also equals 16.

We once again have much more variety

compared to the first deal.

Lastly, you reach the desserts.

Just like with the drinks,

you did not agree to get different ones,

so you can both get ice cream.

If you end up ordering different ones,

it matters to you who gets each one.

For instance, if Amy wants the choco pizza

and you want the brownie bites,

it's crucial who gets each one.

Therefore, we must use

variations with repetition here as well, right?

Well, not really.

Domino's doesn't need to know

who the two desserts are for.

They just put them in the bag

and let you distribute them among yourselves.

In such a case,

you would order the choco pizza and the brownie bites

regardless of who prefers each dessert.

Therefore, we are dealing with combinations with repetition,

instead of variations with repetition.

Recall the formulas from the bonus lecture

on combinations with repetition.

The number of different ways we can use

p elements out of n

with possibly recurring values equals

n plus p minus 1 factorial

over p factorial times n minus 1 factorial.

Since we are picking two desserts out of a possible seven,

we can plug in these values to get 7 plus 2 minus 1,

or 8 factorial,

over 2 factorial times 6 factorial.

After some simplifications,

this equals 7 times 8 over 1 times 2,

or 28.

Therefore, there are 28 different ways

to pick our two desserts out of the menu.

Right, so far, we have found out that we have 325 ways

of choosing our pizzas,

16 options for our drinks,

and 28 combinations of desserts we can order.

Therefore, we need to multiply these

to get a total of 325 times 16 times 28,

or 145,600 different options for this deal.

Whoa, that is a lot of variety!

That is definitely more than 8 times greater

than 728 options you could get with the first deal.

So Amy was wrong after all.

Before you place your order,

you still go through the third deal

to make sure you're choosing the offer

with the greatest variety.

The Double Delight contains

the same number of pizzas and desserts

as the Family Deal.

Therefore, you already know you have 325 ways

of choosing the pizzas

and another 28 ways of picking the desserts.

Thus, you only have to estimate

the number of ways to pick the beverages

that go with your order.

The drinks are large,

so just like with the pizzas,

you decide to get different ones,

since you can share.

Coca-Cola is still running the campaign

where they put names on the labels of each bottle,

so it matters who gets each drink.

For instance, if you want a Coke

and your sister wants a Sprite,

you would not want the Coca-Cola label

to have her name instead of yours.

Therefore, it's important

how you order the two beverages

and the special directions you leave for the delivery.

This suggests you must use variations without repetition

to determine the appropriate number of possibilities.

According to the formula we introduced earlier,

the number of ways of choosing your beverages

would equal 4 factorial over 2 factorial.

This results in you having 12 different ways

of getting your drinks.

Since there are 325 ways of picking the pizzas,

12 ways of choosing the drinks,

and 28 ways of selecting the desserts,

that makes up a total of 109,200

distinct, possible orders you can make.

Just like with the Family deal offer,

the number of possible menus you can get

is more than 8 times greater

than the 728 available for the One Person Combo Pack.

Thus, you showed your friend

that ordering twice as much food

results in having much greater variety

than she had initially thought.

Because 109,200 is less than 145,600,

you decide to go with the Family Deal

instead of the Double Delight

because of the greater variety it provides.

Even though the only difference between the two deals

is the size of one drink,

it results in us having 36,400 more options.

This is a perfect example

of how small changes

can severely expand the possible number of outcomes we have.

This only goes to show

that when using combinatorics,

the devil is in the details,

because every small difference matters.

Great job, everybody!

In this long, practical example,

we manage to go over the various different parts

of combinatorics,

and when we use each one.

We put all the formulas from the section to good use

and made sure we provide logical arguments

before deciding which one to apply.

The goal of this practical example

was to show you how to apply a probabilistic approach

to everyday scenarios

and get into an analytical mindset.

That was also the end of this section.

Next time, we are going to explore a completely new topic

called Bayesian Inference.

Thanks for watching.