Math 123 - Elementary Statistics - Lecture 12
Summary
TLDRIn 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
📚 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.
🔢 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.
🏎️ 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.
🏠 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.
🛣️ 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.
🃏 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.
Mindmap
Keywords
💡Permutation
💡Factorial
💡Zero Factorial
💡Sudoku
💡Permutations of n objects taken r at a time
💡Distinguishable Permutations
💡Combinations
💡Probability
💡Calculator
💡Counting Principles
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
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