Math 123 - Elementary Statistics - Lecture 12

Summary

In this educational video, the focus is on probability and counting, specifically on permutations and combinations. The instructor explains permutations as ordered arrangements of objects, using factorials to calculate the number of permutations. Examples include arranging objects and solving a Sudoku puzzle. The concept of combinations is also discussed, which involves selecting objects without considering order, using a different formula. Practical examples like choosing race car positions and probabilities of card games are used to illustrate these mathematical principles.

Takeaways

• 📚 The section focuses on additional topics in probability and counting, aiming to teach how to arrange objects in order, choose objects without regard to order, and use counting principles to find probabilities.
• 🔢 A permutation is defined as an ordered arrangement of objects, and the number of different permutations of n distinct objects is represented by n factorial, denoted as n!.
• ⚙️ The concept of factorial is explained, which is a product of all positive integers up to a given number, and calculators can be used to compute factorials quickly.
• 💡 An example using a Sudoku puzzle illustrates how to calculate the number of permutations for the first row of a 9x9 grid, which is 9 factorial.
• 🔄 The formula for permutations of n objects taken r at a time is introduced as nPr = n! / (n-r)!, which is used when the order of arrangement matters.
• 🏁 The concept of distinguishable permutations is discussed, which accounts for arrangements where objects of the same type are indistinguishable, using the formula n! / (N1! * N2! * ... * Nk!).
• 🎯 An example of distinguishable permutations is given with a building contractor arranging different types of houses in a subdivision.
• 🎲 The section transitions to combinations, which are selections of objects without regard to order, using the formula nCr = n! / (n-r)! * r!.
• 🚧 An example of combinations is provided with a State's Department of Transportation selecting companies for a highway project from a set of bids.
• 🃏 A probability example is discussed, calculating the likelihood of being dealt five diamonds from a standard deck of 52 playing cards, emphasizing the irrelevance of order in such cases.

Q & A

What is the main focus of section 3.4 in the transcript?

The main focus of section 3.4 is on additional topics in probability and counting, including how to find the number of ways to arrange a group of objects in order, how to choose several objects from a group without regard to order, and how to use counting principles to find probabilities.

What is the definition of permutation as explained in the transcript?

Permutation is defined as an ordered arrangement of objects where the order is important. It refers to the number of different ordered arrangements that can be made from a set of distinct objects.

How is n factorial calculated and what does it represent?

n factorial (denoted as n!) is calculated by multiplying all positive integers from n down to 1. It represents the number of different permutations of n distinct objects.

What is the significance of the exclamation point in mathematics as mentioned in the transcript?

In mathematics, the exclamation point after a number (e.g., n!) denotes the factorial of that number, which is the product of all positive integers less than or equal to that number.

How does the transcript describe the process of calculating factorial using a TI-84 calculator?

The transcript describes calculating factorial on a TI-84 calculator by pressing the 'math' button, navigating to the probability section using the right arrow, and selecting the fourth item listed, which is the factorial function.

What is the value of zero factorial as stated in the transcript?

Zero factorial (0!) is defined to be one in the transcript.

How many different ways can the first row of a blank 9x9 Sudoku grid be filled according to the transcript?

According to the transcript, the first row of a blank 9x9 Sudoku grid can be filled in 362,880 different ways, which is 9 factorial.

What is the formula for calculating permutations of n objects taken r at a time as mentioned in the transcript?

The formula for calculating permutations of n objects taken r at a time is nPr = n! / (n-r)!, where n! is n factorial and (n-r)! is the factorial of (n minus r).

How does the transcript explain the concept of distinguishable permutations?

Distinguishable permutations are explained as the number of ways to arrange n objects where there are N1 objects of one type, N2 objects of another type, and so on. The formula for this is n! / (N1! * N2! * ... * Nk!), where the sum of Ni values equals n.

What is the difference between permutations and combinations as discussed in the transcript?

Permutations involve ordered arrangements of objects where the order matters, while combinations involve selections of objects without regard to order. The formulas for calculating them differ, with permutations using nPr = n! / (n-r)! and combinations using nCr = n! / [(n-r)! * r!].

How does the transcript illustrate the use of combinations in a real-world scenario?

The transcript uses the example of a State's Department of Transportation needing to select 4 companies from 16 bidding companies for a project. The number of different combinations of 4 companies that can be selected is calculated using the combination formula.

Outlines

00:00

📚 Introduction to Probability and Counting

The speaker introduces the section on additional topics in probability and counting, focusing on permutations and combinations. The objective is to understand how to arrange objects in order (permutations) and to choose objects without regard to order (combinations), and to apply counting principles to find probabilities. The concept of permutation is defined as an ordered arrangement of objects, emphasizing the importance of order. The mathematical representation of permutations is introduced as n factorial, which is a product of all positive integers up to n. The speaker provides an example of calculating 6 factorial and discusses how to use a calculator, specifically a TI-84, to find factorial values. The concept of zero factorial is also mentioned, stating it is defined as one.

05:01

🔢 Permutations and Factorials Explained

The speaker delves into permutations with an example of a Sudoku puzzle, illustrating how to calculate the number of ways to fill the first row of a 9x9 Sudoku grid. The calculation involves 9 factorial, which is the product of all numbers from 1 to 9. The speaker demonstrates how to use a calculator to find 9 factorial and arrives at the result of 362,880 possible ways to fill the row. The concept of permutations of n objects taken r at a time is introduced with a formula, nPr, which is n factorial divided by (n - r) factorial. An example of forming a four-digit code without repeating digits is used to explain the formula, resulting in 5,040 possible combinations.

10:01

🏎️ Permutations in Racing and Counting Codes

The speaker explores permutations further with an example of the Indianapolis 500, where 33 race cars compete. The task is to determine how many ways the top three finishers can be arranged. Using the formula for permutations of n objects taken r at a time, the calculation results in 32,000,736 possible outcomes. The concept of distinguishable permutations is introduced, where objects of different types are arranged. An example of a building contractor planning a subdivision with different types of houses is used to explain this concept, leading to 13,860 distinguishable arrangements.

15:03

🏠 Distinguishable Permutations in Real Estate

The speaker continues with the topic of distinguishable permutations, using the example of a building contractor arranging houses of different types in a subdivision. The formula for this scenario involves dividing the total number of permutations by the factorial of the number of each type of house. The calculation is demonstrated, and the speaker emphasizes the importance of using parentheses correctly when entering the formula into a calculator. The result is 13,860 distinguishable ways to arrange the houses.

20:06

🛣️ Combinations and Highway Bidding

The focus shifts to combinations, where the order of selection does not matter. The speaker uses the example of a State's Department of Transportation selecting four companies from 16 bids for a highway project. The formula for combinations, denoted as nCr, is introduced, which is similar to the permutations formula but without multiplying by r factorial in the denominator. The calculation results in 1,820 different combinations of selecting four companies from 16.

25:06

🃏 Probability of Selecting Board Members and Card Hands

The speaker concludes with examples involving probability. The first example calculates the probability of randomly selecting three members for a student advisory board from 17 members, where order matters. The calculation results in a probability of approximately 0.0002. The second example calculates the probability of being dealt five diamonds from a standard deck of 52 cards, where order does not matter. The probability is found to be 0.0005. The speaker emphasizes the low likelihood of such events and concludes the chapter with these examples.

Keywords

💡Permutation

Permutation refers to an ordered arrangement of objects. In the context of the video, permutations are used to calculate the number of distinct ways objects can be arranged. For instance, the video explains that the number of different permutations of n distinct objects is calculated using n factorial, which is a product of all positive integers up to n. The concept is used to solve problems such as arranging the first row of a Sudoku puzzle, where the order of numbers matters.

💡Factorial

Factorial, denoted by an exclamation point (n!), is a mathematical operation that multiplies a number by all positive integers less than itself down to 1. In the video, factorial is used to calculate permutations, with the example of 6! being calculated as 6 × 5 × 4 × 3 × 2 × 1, which equals 720. The concept is crucial for understanding how to determine the number of ways to arrange objects in a specific order.

💡Zero Factorial

Zero factorial is a special case in mathematics where 0! is defined as 1. The video mentions this as a given, emphasizing that even though it might seem counterintuitive, it is a standard definition used in combinatorics. This concept is important for understanding the boundaries of factorial calculations and their applications in counting problems.

💡Sudoku

Sudoku is a number puzzle where the objective is to fill a 9x9 grid such that each row, column, and 3x3 subgrid contains the digits from 1 to 9 without repetition. The video uses Sudoku as an example to explain permutations, specifically calculating the number of ways to fill the first row of a 9x9 Sudoku grid, which is done by calculating 9 factorial.

💡Permutations of n objects taken r at a time

This concept refers to the number of different permutations that can be formed from n distinct objects by taking r of them at a time. The video introduces a formula for this, nPr, which is calculated as n! divided by (n-r)!. This is used to solve problems such as creating a four-digit code with no repeated digits, where the formula helps determine the number of possible codes.

💡Distinguishable Permutations

Distinguishable permutations involve arranging n objects where there are repetitions of certain objects. The formula for this, n! divided by the product of factorials of the number of repetitions (N1! * N2! * ... * Nk!), is used when the objects being arranged are not all unique. An example from the video is planning a subdivision with different types of houses, where the formula is used to calculate the number of distinguishable ways houses can be arranged.

💡Combinations

Combinations refer to the selection of objects from a group where order does not matter. The video explains combinations using the formula nCr, which is n! divided by (n-r)! * r!. This is used to solve problems such as selecting four companies from a group of sixteen, where the order of selection is irrelevant.

💡Probability

Probability in the video is discussed in the context of calculating the likelihood of certain outcomes based on permutations and combinations. For example, the probability of selecting three specific members for a board's chair, secretary, and webmaster positions is calculated by dividing the number of favorable outcomes (one correct selection) by the total number of possible selections (17P3).

💡Calculator

The video provides guidance on using a calculator, specifically a TI-84, to compute factorials quickly. This is an important practical skill for students as it allows them to calculate large factorials that are often used in permutations and combinations problems. The video demonstrates how to navigate the calculator's menu to find the 'prob' section where the factorial function is located.

💡Counting Principles

Counting principles are foundational in combinatorics and are used throughout the video to solve problems involving permutations and combinations. These principles help in determining the total number of outcomes for a series of events, which is crucial for understanding probabilities. The video uses counting principles to explain how to find probabilities in various scenarios, such as being dealt five diamonds from a standard deck of cards.

Highlights

Introduction to section 3.4 focusing on additional topics in probability and counting.

Objectives include finding arrangements of objects, choosing objects without regard to order, and using counting principles for probabilities.

Definition of permutation as an ordered arrangement of objects where order matters.

Explanation of n factorial, which represents the product of all positive integers up to n.

Example calculation of 6 factorial to illustrate the concept of factorials.

Instruction on how to use a TI-84 calculator to find factorials.

Zero factorial is defined as one.

Example of calculating permutations using a Sudoku puzzle.

Formula for permutations of n objects taken r at a time, denoted as nPr.

Explanation of how to calculate permutations of objects taken r at a time with an example of a four-digit code.

Example of calculating permutations for the Indianapolis 500 race finishers.

Introduction to distinguishable permutations and their formula.

Example of arranging houses in a subdivision using distinguishable permutations.

Definition and formula for combinations of n objects taken r at a time.

Example of selecting companies for a highway project using combinations.

Calculating the probability of selecting members for a student Advisory Board.

Calculating the probability of being dealt five diamonds from a standard deck of cards.

Conclusion of the section and chapter with a summary of key points.

Transcripts

hey everyone welcome back we are in

section 3.4 today we are looking at

additional topics in probability and

Counting

so our objectives for this section are

how to find the number of ways a group

of objects can be arranged in order how

to find the number of ways to choose

several objects from a group without

regard to order and then how to use the

counting principles to find

probabilities

so we'll begin with a definition

definition is for permutation so

permutation is an ordered arrangement of

objects so that's the the main thing

that you need to understand

when you're when you're looking at

permutation when you hear that word you

have to remember that permutation really

is is about dealing with order

all right so it's the number of

different or the number of different

permutations of of n distinct objects so

if we have a bunch of different objects

and we want to know what are all the

different uh ordered Arrangements that

we can put them in

the way that we can sort that out is

like so now this doesn't mean that

you're going to scream the letter N

um but rather this is what we call n

factorial

so in mathematics the exclamation point

actually means something and it's the

word factorial so what n factorial or

anything with a factorial means is it's

it's a kind of multiplication so in

general terms

what n factorial means is this it says

you take whatever the value of n is

and then you multiply that by the number

that is one less so 1 less than n would

be n minus one and then you multiply

that by one less so then you'd be at

n minus 2

and then you multiply that by one less

and you'd be at n minus 3 and you get

the idea and you keep going right and so

and you keep multiplying until you

eventually find your way

to one and once you make it all the way

down to one then you stop and you have

have your solution

so as just sort of like maybe an example

to kind of understand this idea

if I said you know what is

6 factorial

well 6 factorial would be six times five

times four times three times two times

one and it's not always necessary to

write it all the way out like this in

case you're wondering now every

every time I have factorial do I have to

write all the numbers out no not always

sometimes it's helpful to do it because

there will be cancellations that you'll

be able to make when you can see them

written out

um but for the most part no you're not

going to have to write them out but this

is just so you get an idea of what it

means so if you were to make all these

multiplications this one would come out

to 720

now there's a couple ways you could make

the multiplication you could sit there

and just one by one go through and do it

or you could use your calculator to do

it

if you have the TI-84 calculator that

was recommended for this class

I can just kind of give you a brief

Direction on how to find that factorial

button so that you can use it quickly on

the left side of your calculator

directly underneath the alpha key the

alpha key I think is colored in green

there is a math button

if you push that math button you're

going to get a display a menu that's

going to come up and what you're going

to do is you're going to use the right

arrow and you're going to go over

to the probability section

and it's going to be labeled as prob

and once you're there you're going to

see it immediately it's the fourth

fourth item listed

all right there may be some other ways

to access the factorial on the

calculator but this is the most direct

way that I'm aware of I don't know if

some of them have it

um

directly on them depending on the

calculator you have I'm not familiar

with it being a simple just push second

and a button and can access it but this

is the way that that I use it so that's

what I know if you know another way

you're welcome to share that with me I'd

love to know so anyway

um and then just as sort of uh maybe not

so much an example but just a note

just so we're aware

if you should encounter

zero factorial

we count zero factorial as one when we

use zero factorial okay so that's just

more of a given you can think of that as

like just like a separate definition

that zero factorial is given to be one

okay all right so let's let's dive in

and let's let's look at an example

so

many of you know what this is this is a

Sudoku puzzle

um these were the things that uh

kept me busy in all my non-math classes

when I was at Cal Poly they used to have

one and we had a school paper that came

out every day and in it they had a

crossword puzzle and a Sudoku puzzle I

made sure to grab a copy of the paper

every day and anytime I went to a

non-math related class boy you better

believe that that's what I was doing

although I probably shouldn't be

advocating that to you now that I think

about that because you probably feel the

same way about this class that I did

about my non-math classes so you know

what forget that whole idea never mind

ignore that I'm gonna try to edit that

out

um okay so example finding the number of

permutations of n objects okay so the

objective of a nine by nine Sudoku

number puzzle is to fill the grid so

that each row each column and each three

by three grid contain the digits one

through nine

how many different ways can the first

row

of a blank blank 9x9 Sudoku grid be

filled up all right so assuming that

there are no numbers in the puzzle to

start which would be a really

annoying puzzle to deal with I imagine

um

the number of permutations that you

would have could be found

by 9 factorial

so 9 factorial we know what factorials

are now that's nine times eight times

seven times six times five times four

times three times two times one and

again no you don't have to write it all

the way out every single time but just

until we really kind of get a good grasp

on it really want to hammer home what

factorials are so that's why I'm writing

it out here

if we use the calculator though and type

in 9 and then use the directions up

above for how to find factorial

let's just do that real quick here so

I've got my calculator handy

so let me go to nine and then math over

the probability down to option four nine

factorial

there are three hundred sixty two

thousand

eight hundred and eighty

ways for this to happen all right so

that's it's a lot of different a lot of

different options for how to fill that

in

okay

so what about permutations of objects

taken are at a time what does that mean

permutations of n objects taken r at a

time well this is the number of

different permutations of n distinct

objects taken r at a time this is given

by a formula

that we're going to denote n PR and it's

going to look like this with the with

the letters being little the n and the r

being small on either side of the p and

we're going to

identify this formula as n factorial

divided by

n minus r

factorial so that just means with the

parentheses there we'll make the

subtraction first and then we'll apply

factorial to that number and this is

going to be true where

R is given to be less than or equal to n

okay so that yeah that's super clearish

right clear as mud as they say

um

wait what what what am I talking about

what is this guy going on about right

now

yeah I get it this this looks very

bizarre and very weird

let me explain it to you an example I

know a lot of this stuff comes at you in

these abstract ways and it may seem a

little bit confusing but

um let me explain it with some specifics

and then hopefully it'll make sense for

us so using this formula we want to find

the number of ways of forming a four

digit code in which no digit is repeated

all right so

in this situation

um we're going to need to select let's

say four digits

from a group of

10. Now where's the group of 10 coming

from well the numbers 0 through 9. right

so we we can pick any number zero

through nine which would

amount to ten separate digits so in this

case we have n is equal to 10 those are

all the options we have to choose from

and we're picking four

all right so the way that the formula

would be set up is we would have 10

p 4

so that would be equal to 10 factorial

over

10 minus 4 factorial

or this would be 10 factorial

divided by 6 factorial all right now one

of the things that you cannot do just so

that we're all clear you cannot just

cancel factorials so this would not just

be 10 over 6.

you also cannot simply just reduce 10

and 6 by themselves

but there is a way to do some

simplifying and let me show you how to

do that

so 10 factorial is 10

times 9 times 8. times 7

times now I could keep going but maybe

I'm maybe let's say I don't I don't want

to keep going with this and I just want

to stop here

it's 6 all the way down to one right

well 6 all the way down to one

is 6 factorial

over

6 factorial

these can cancel I am allowed to cancel

6 factorial because it's the it's the

exact same value so those values can be

eliminated so then all I'm left sorting

out here is what is 10 times 9 times 8

times 7.

and if I make those multiplications I

get five thousand

forty so there are 5040 ways

of forming a four-digit code in which no

digit gets repeated

okay all right so there's

obviously there's there's other ways

that you could do this right you could

sit sit down and just start listing out

four digit options

uh where you're not repeating anything

but think about how much time that would

take you like if you just started

listing out all the possible four digit

combinations for a code where you're not

repeating the numbers that's going to

take a long time so this this gives you

um

a very concise clean neat approach with

a nice formula for how to sort these

events out

okay

let's look at another one

so every year

33 race cars are in the Indianapolis 500

how many ways can the cars finish first

second and third so another in other

words

of the 33 race cars that start the race

how many different ways how many

different options are there to have uh

first and second and third sorted out

so in this case we are going to select

three cars from our group of

33

. so here n would be 33.

and R is 3. so we're going to do 33

p

3

and using our formula this would be 33

factorial

divided by

33 minus 3

factorial

so making the subtraction we have 33

factorial over

30 factorial and if we use a similar

approach that we had in the last example

we can start to list out 33 factorial as

33

times 32 times 31

and then everything after 31 would just

be 30 all the way down to 1 which could

be expressed as 30 factorial

and then all over 30 factorial and we do

it that way so that way we can make this

cancellation

so cancel your 30 factorials and so now

all you're multiplying is 33 times 32

times 31

which comes out to 32 000

736.

so that's all the different ways

that the 33 Race Cars could be arranged

to finish first second and third in this

particular race so that's just that's a

lot a lot of different options to look

at there

okay so hopefully we're getting the idea

here

all right let's look at what's called

distinguishable uh permutations now

so with distinguishable permutations

this is the number says the number of

distinguishable permutations of n

objects where N1 are of one type and two

are of another type and so on this is

going to be defined as such this is

going to be n factorial

divided by

we're going to have N1 factorial times

and 2 factorial

times and on and on and on and on

and all the way down to wherever we end

and for now we'll just call that N Sub K

factorial now it looks a lot worse than

it is and let me just explain to you

what

what all these numbers in the bottom

mean we're going to say where

N1

plus N2 plus all the way up to N Sub K

if we were to add all these things up in

the denominator it would just total up

to the N that is in the numerator now

again in the abstract this may seem very

confusing

but in the context of a specific example

yeah I don't think this is going to be

that bad I think it's it's going to be

pretty easy to sort out so let's do that

let's look at this in terms of an

example

all right so a building contractor is

planning to develop a subdivision now

the subdivision is going to consist of

six one-story houses for two-story

houses and then two split level houses

in how many distinguishable ways

can the houses be arranged well you can

already kind of see it the the little

graphic in the bottom right corner gives

it away but pretend you didn't see that

already

um there are 12 houses total in the

subdivision all right so that's our n

value that's 12.

and then n one and two and N3 those

represent the different types of houses

that are going in the subdivision so for

example N1 could represent the one-story

houses there are six of those

and two would represent let's say the

two-story houses we're building four of

those and then in and three those are

the the split level houses and there are

two of those

okay so if we want to know how are their

or how many distinguishable ways rather

can these houses be arranged well let's

just use the formula so it's going to be

n factorial

over

and 1 factorial times

and 2 factorial times N3 factorial

so this is going to be 12 factorial

over

6 factorial

times 4 factorial times

2 factorial

now we could work it out to where we do

a little canceling if we were to take 12

factorial and start to expand it out we

could cancel off any one of the six or

the four or the two we couldn't cancel

all of them we could cancel one of them

um but you as you can see from the

graphic on the right we could also just

enter this whole thing into the

calculator and just have it sort the

whole thing out notice the only thing to

be careful of though is the use of

parentheses you need to include those in

the denominator otherwise your

calculator will not give you

the the answer you're looking for here

so we can see that there are going to be

13

860

distinguishable ways that these houses

can be arranged uh in this subdivision

okay so there we go

all right let's talk about combinations

now so combinations of n objects taken r

at a time well this is a selection of

our objects from a group of n objects

and here's the important thing this is

without

regard to order okay so when we were

talking about permutations order matter

the order was real important how how

things were being placed

here we don't care about the order here

we just want to know how many how many

different ways can we can we group the

things together without really caring

how they're

how they're ordered up

all right now the way we're going to

sort this out is with a different

formula

so for a combination we'll use C so it's

NCR and this is found it's very similar

to the other one it's n factorial

over

n minus r factorial only difference is

here

where the denominator is being

multiplied by R factorial and again

we know that R is going to be less than

or equal to n

all right so this is the formula for

dealing with a combination

all right well let's look at an example

so a State's Department of

Transportation plans to develop a new

section of Interstate Highway and they

receive 16 bids for the project

the state plans to hire four of the

bidding companies how many different

combinations of four companies can be

selected from the 16-bidding companies

all right well let's look at this we

need to pick four companies from the

group of 16. now we're not picking them

in any particular order

we just want to know how many different

ways can we select four out of these 16.

okay so here n is going to be 16. and R

is going to be 4 and again order is not

important

okay so let's sort it out so we're going

to have 16

C 4 and this is going to be 16 factorial

over

16 minus 4 factorial

and then times 4 factorial

so if we make the subtraction we get 16

factorial over

12 factorial times 4 factorial

and what we'll go ahead and do here is

we'll do what we've done previously and

look to simplify a little bit so with 16

factorial that's 16 times 15 times 14.

I'm going to take this all the way down

to 12.

and I'm going to stop it at 12 and then

leave the factorial to to represent 12

all the way down to 1 and then this is

over

12 factorial times 4 factorial and the

purpose there of course is so we can do

a little canceling we can get rid of

these 12 factorials which just

simplifies the process a little bit for

us

so what we end up with is 16 times 15

times 14 times 13

and all of that is over 4 factorial

and so we can enter that into our

calculators

and we end up with 1 820

different combinations

so there are 1820 different combinations

of selecting four of those bidding

contracts out of 16.

total bitters

okay so pretty similar to the

permutation stuff not that different

all right well in the last set of notes

if you remember

um I left in there a section talking

about

the all the different types of

probabilities and some of the different

formulas you were dealing with because

things were just maybe getting a little

overwhelming or confusing and so with

this new information in terms of like

permutations and combine combinations

and these formulas again it can be a

little bit much and so if if you found

that other page to be useful you might

find this useful as well so feel free to

go ahead and grab this and print it or

uh scan it or do whatever with it and

include it with your notes or with your

study materials just to kind of help you

work through this chapter

all right let's do another example

so a student Advisory Board consists of

17 members three members serve as the

board's chair secretary and webmaster

each member is equally likely to serve

any of the positions

what is the probability of randomly

selecting the three members who will be

chosen for the board okay

so in this instance order is going to be

important

and there's only going to be one

favorable outcome only one scenario is

going to yield what we want and only one

one event are we going to pick the right

person who is the president the right

person who is

I've already forgotten the positions

um the person or right not the president

chair the secretary and the webmaster

right we can only find them in one

combination so how are we going to do

this well first let's figure how many

how many different ways are there to

pick

so this is going to be

17 people

for three spots

so we would use 17 factorial

over

17 minus 3 factorial

which is 17 factorial over 14 factorial

and if we simplify it out again we can

just break down the numerator a little

bit here

and we can make our cancellations of the

14 factorial so what's 17 times 16 times

15 that's 4080.

so this represents all the different

ways that we could grab three people out

of 17 and put them in a specific order

all right now what we want to know is

what is the probability

of selecting

the three members

now remember there's only one right way

to do this

so there is one way to select out of

4080 possible ways of doing this so one

out of 4080 or roughly 0.0002

so pretty slim chance

that you could walk into a room of 17

folks and pick the exact three in the

exact order

um that you would need

okay let's do another one

let's find the probability of being

dealt five diamonds from a standard deck

of 52 playing cards okay

so the way we're going to sort this one

out is to know in a standard deck of

playing cards remember 13 of the 52

cards are diamonds now note that it does

not matter what order the cards are

selected okay so here order is

irrelevant as long as they're diamonds

that's fine the order that they are

appearing is it it doesn't doesn't take

any precedent

the possible number of ways of choosing

five diamonds out of the 13 diamonds in

the deck is 13 C5

the number of possible Five Card hands

in the entire deck is 52 C5

so what we need to do to figure out this

probability

is we have to take the probability

of getting five diamonds

dealt to us

would be

13 C5 right the probability of getting a

five card Diamond hand out of

all the possible Five Card hands that

could exist okay well

do we know how to sort these out right

we've done enough of these I think at

this point to know how these work so if

you want to go through and do the Nitty

Gritty with all the factorials you can

go right on ahead let me give you the

values and maybe this would be a good

opportunity for you to do a little

practice and see if you can get that

right value

so of getting five diamonds

you're going to get 1287

and then just getting

um a five card hand all the different

combinations of Five Card hands that

exist out of a 52 card deck there are

two million

598 960.

do you think there would be that many

different

combinations of Five Card hands that you

could get in a deck of cards well

if we make the division here's our

probability

it's going to be 0.0005

so basically

um

don't go to the casinos counting on

getting a flush if you're a poker player

all right we're gonna stop there

that does it for this section in fact

that does it for this chapter so uh I'll

see you again in the next chapter