Permutations & Combinations S1 Intro | Factorials, Restrictions, Arrangements

everyone I hope you guys are doing well so finally I am starting a video series on permutations and combinations of a

level math S1 so what we've done is my team and I we have gone over all the pass questions and we have divided

permutations and combinations into multiple categories so what I'll do is first I will this is going to be an

introductory video by the way but after that I will first cover all of permutations I've divided permutations

into seven different parts okay you can you can check out the seven parts I'll leave a link in the description where

you can download the PDF you'll find all the P papers with answers and uh you'll see perations and combinations divide

into different categories so we will be solving possible questions of every single category and uh that way we can

understand how permutation combination works and the type of questions that you will be tested on this video however is

just going to be an introductory video so yeah without wasting any more time let's start so permutations and

combination so if you if you have done adms in your igcc or o levels you have some idea but U the questions that we

have in S1 are actually far more difficult than the ones we have in adms okay so first let's understand the

difference between the two okay and then like I said we're going to study permutation in detail so let's first

talk about the difference between the two so starting with permutation now remember permutation and combination

both have their separate applications and permutations is basically used when we talk about permutation so this is

basically used when you are selecting and arranging okay so a lot of people say that we use this only when we're

arranging well that is true but before you can arrange something you have to select what you have to arrange okay so

when you take when we're talking about permutations that is when you have different objects or even you have

multiple objects and uh there are some objects that are similar and you want to arrange them so obviously you're going

to first select the ones that you have to arrange and then you're going to arrange them okay so that is where we

use permutations now this can be applied to objects okay this can be applied to

objects like books uh pictures and uh what else books pictures

um okay I can't really think of think of anything else but uh you you don't really there's no there's no restriction

as to where we can apply them this is uh this this can be applied to objects it can be it can also be applied to people

okay you can use permutations to arrange people in a straight line it can also be applied to

letters all right and it can also be applied to numbers now in all these remember that

since we are arranging them so whenever you're arranging something the order in which you arrange matters a lot okay so

here remember that the order is important all right now to give you an idea of how permutations basically work

and what exactly do I mean by order now let's say there are three people who have participated in a contest and there

are two judges okay so let's say there is Judge one judge one and judge 2 okay now there

are the three people who have participated or the three people who have been shortlisted for the podium are

that which basically means first second third are a b and c so judge one says that as for my judgment I think a should

get the first place C should get the second place and B should get the third place okay so this is the ranking order

as per the first judge now judge two on the other hand says that nope I think B should get the first place a should get

the second place and C should get the third place now because we have rankings attached to these people so which means

the order is important so can we say Well they're both the same I mean both the judges have the same have

shortlisted the same three people so it doesn't really matter who gets first second and third can we say that of

course we can't why because here the order is important why because we're arranging them we're arranging first

second and third place so that is why the order is important so another example would be let's say that you have

a four-digit password to your iPhone okay let's say the password is 2357 so if somebody just tells you the

numbers to the password like the numbers are 5327 is that enough can you then well technically after multiple tries

you can figure out what the correct password is but the thing is you can't just enter the correct digits in

whatever order and expect to unlock the font you have to get the order right as well so here technically the concept

that we're using is permutations why because changing the order leads to a different password all right so remember

long story short in permutations AB that is arranging two objects A and B as AB is not the same as ba okay but when we

are talking about combinations on the other hand so in combinations what we're doing is we're just selecting only we're

not arranging them we're only concerned with selecting and this is usually done when you are making a team okay why

because if you're making a team then in that case the order in which you make the team is not important the order is

not important all right and in the case of combinations AB that is two people who

are A and B respectively is the same as ba why because these are the same people so let's say somebody says that you know

what I want two players for the upcoming table tennis tournament and the players that I choose are ab and then somebody

else comes and says that you know what I don't like the people that you've chosen I want ba well does that really make

sense no it doesn't so should should there be a debate over this no because it's a team it It ultimately doesn't

matter however if you're making a team but within the the team there is a certain Merit like for example who's

going to be Captain who's going to be Vice Captain so let's say if here they're not just making a team but

they're also deciding who's going to be captain and who's going to be Vice Captain then a being captain and B being

what the Vice Captain is not the same as B being Captain a being the Vice Captain okay so whenever you have some sort of

Merit some sort of ranking that's involved and you're making a team then in that case we use permutation okay but

if there is no merit no ranking so that is when we use combination so coming back since these are the same people

coming back to the point since these are the same people therefore it's considered the same team okay now what

we're going to do is we're going to put permutation and combination we're mainly combination on the back burner for now

and we're going to focus on a prerequisite of permutation which is basically called factorial notation and

that is this factorial key that you will have on your calculator so if you press shift and if you press uh the button

that's right next to sign you will see an exclamation Mark now that's not exactly an exclamation mark it's called

factorial and we'll understand how it works so let's say here's a question which we can solve through factorial

notation and not just through factorial notation we can also solve it through uh permutation okay so we're going to

understand how they're both kind of the same and used interchangeably so it says how many ways can three people be

arranged in a straight line now notice the keyword here is arrange so what we have to do is we have to arrange them

and whenever we arranging people the order in which we we arrange them is important okay so a good way to start

would be now if you know the concept already you probably have figured out the answer immediately but let's let's

manually find out the number of a that actually we can arrange them so I'm going to call one person a the other B

and the third person C okay so let's make three blanks and let's write down one by one all the possible ways in

which we can arrange them so one obvious way is obviously a b c but is this is this the only way that I can arrange

these three people of course course not another would be let's say a c b so I've kept a in the same place but I've

shuffled B and C another would be let's say we keep um B in the first place and then a in the first and C in the a in

the second and C in the third now let's see if there is another possibility of course there is it could be b c a now

what if C wants to be in the first place so in that case we would have c a b and notice that we can Shuffle A and B so

that would make it C B A okay now let's count 1 2 3 4 56 now I encourage you to think of any other possibility and with

a bit of thought you'll realize that there isn't so altoe you can see that we have how many ways that we have six

different ways in which three people can be arranged in straight line so does that mean that every time you have a

question like this we have to do it like this of course not there is an easier way so the easier way is the shortcut is

that you make make three blanks since we have to arrange three people 1 2 3 now for the first place we have three

options because nobody has been arranged yet all three people are available to be arranged that means in the first place

we have three options now in the second place since one person whoever it may be a b c has been arranged or has taken a

seat now we have to look for now we have to look for the second place for which we have two options and then lastly for

the last place we have one option so that means all together all I have to do is simply multiply them so that would be

3 * 2 * 1 which is equal to 6 okay now this is also called the basic counting principle that is if you can do task a m

number of a and task two n number of a then together these two tasks can be performed M into n number of ways okay

just like if you remember going back uh in all levels if let's say you roll two dice and you want to find out the total

number of outcomes so since there are six outcomes six different outcomes in die one and six different outcomes in

die two together you have 6 * 6 which is 36 outcomes similarly if let's say you toss a coin and you roll a die a fair

six-sided die then altogether you would have 12 different outcomes why because two in the coin and six in the die so 2

into 6 is equal to 12 so this is kind of like basic counting principle also but the reason why we have termed this as

factorial notation is because 3 into 2 into 1 is basically the same as 3 factorial which is 3 into 2 into 1 and

that's equal to what that's equal to six ways okay now or you may be thinking that where does permutation fit into

this so this can be done using permutation as well so for permutation the key that we use is p which you will

find on your calculator if you press shift and then if you press the multiplication key so I don't I don't

think if this is in Focus but you basically see a p on your calculator now how do we use this or how can we use

permutation over here very simple how many people do we have available that are to be arranged that's three and how

many people do we have to arrange of the three we have to rrange all three of them so that would be 3 P3 which is the

same as 3 factorial or 3 into 2 into 1 and that's equal to what that is also equal to 6 okay so this is what

factorial notation is we'll write down formally what it is as in what the definition is but first let's understand

its application further and let's do some more examples so here's a question which says how many ways can five people

be arranged in a straight line so here we're talking about five people so with five we can't do it manually the way we

did it with three so instead we are going use a shortcut so 1 2 3 4 5 1 2 3 4 5 yeah so for the first place we have

five options for the second four for the third three for the fourth two and for the fifth and final place we have one

option so that means all together that's either equal to 5 factorial which is 5 into 4 into 1 or 5 P5 which is equal to

well you can work this out you'll notice that it's 120 and you can see why we didn't do it manually because had we

done it manually we I would have only been able to solve this question in this video okay now Part B Part B says how

many ways can three or five people be arranged in a straight line okay so now we have uh somewhat of a restriction

okay the Restriction is that I don't have to arrange all five of them I only have to arrange three of the five people

so that means in the first place I have five options in the second place I have four options in the third place I have

three options so Al together can I say five factorial nope obviously not it's 5 into 4 into 3 5 into 4 into 3 or we had

five people available to be arranged and of these five people we only had to arrange three so that means we can do 5

P3 let's see what the answer to that is so 5 P3 is equals to 60 so that means if you want to arrange three people out of

five you can do that through 60 different ways okay now what is it what what is factorial by definition here as

you can see so I'll write down over here that this is the definition definition okay product of

all positive integers less than or equal to n where N is a positive integer so that means if you have n factorial you

would do n into nus1 into n minus 2 and you'd keep on doing this you'd keep on doing this till

you get 3 into 2 into 1 okay so let's we start from n and we keep on going till the very last positive integer which is

one that is if it's n factorial and what does it represent it represents the number of ways in different objects can

be arranged okay now let's do some examples and we saw that how permutation was applied to arranging uh people now

let's see how it can be applied to arranging letters okay I want you to observe something that's very important

over here so it says how many ways can the three letters of the following words be arranged now in the first part it's

Ali how many letters do we have in the word Ali three letters so that means you can list down all three different ways

or you can just do three factorial why because for the first place you have three options for the second place you

have two and for the third you have one so 3 into 2 into 1 which is equal to what which is equal to six ways okay now

next we have the letter the word Bob now notice that there's something interesting with the word Bob and that

is you have two letters of the same kind okay so how does that work what do we do when we have two letters of the same

kind well let's start by making three boxes and what if I do 3 into 2 into 1 using the same principle as we did with

Ali and say that we have six different ways in which the word pop can be arranged now let's see if we can list

those six ways okay so I want you to pay close attention so I'm going to make six blanks 1 2 3 oh sorry not six blanks I'm

going to make three blanks six times 1 2 3 1 2 3 1 2 3 so that's five 1 2 3 okay now let's start with the one Arrangement

that we can see already that's b o b now what if what if I I swap the bees okay I move this be here and this B here and

say you know what here's another Arrangement and that is Bob now what would you say you would say that wait a

minute this is not exactly the same Arrangement it's the same okay so we will count this as one so even though

we've swapped the B's but it's Bob or still Bob and it's still Bob and let's write down another Arrangement which is

BB o since it's different from Bob that means this will be counted but what if once again I swap the two B's and I say

that you know what here's another Arrangement BB O So once again this these two are one and the same so that

means this is the these two basically are together is another unique way of ranging these three letters now what if

I say that we can do OB BB and then once again I so the B so You' barely notice the difference so that means these two

are the same and altogether there are how many ways there are three ways in which the word Bob in which the letters

of the word Bob can be ranged now the question is how do we do this without actually listing it down well very

simple all you do is in the numerator 3 factorial okay but in the denominator what you do is since there are B's that

are being repeated so you divide it by 2 factorial why are we dividing it by 2 factorial because 2 factorial represents

the factorial of the

number of same objects it represents the factorial of the number of same objects okay so this brings me to a very

important note also now let's make some room here oops Yeah so here's the note I want you

to pay close attention the note is that when letters

slobs when letter slobs are repeated when letters or objects are

repeated we divide by the

factorial we divide by the factorial of the number of

times they are repeated okay now I'd like to show you the general way of doing this okay so let's

generalize this and for that we can write down the official definition here it is so it says the number of distinct

Arrangements of n objects in a line of which P are identical to each other Q are identical to each other R of a third

type are identical to each other is n factorial divide P factorial divide by into Q factorial into R factorial okay

now now let's apply the same concept to the letter uh to the words to the letter in the word trigonometry okay so how

many letters do we have all together we have 1 2 3 4 5 6 7 8 9 10 11 12 so that's that would be 12 factorial but

what if I leave it there obviously that'll be wrong because there are so many letters that are being repeated so

we can see that the the letter the word t OCC occurs twice okay so where just what just happened I have no idea anyway

so we can see that the letter t occurs twice so that that means we have to divide by two factorial so this two

factorial is representing the number of times the letter T has been repeated now o in fact R is also occurring twice so

that means we have to divide by two factorial so this 2 factorial is basically for the two RS and O also is

another letter that is being repeated so that means we have to divide it by two factorial as well and then if you work

this out you will get a huge answer and normally you do get some really huge and crazy values in permutations and combin

so that's perfectly all right don't freak out it's 59 m875 1200 okay so this was just a

concept where you have objects that are being repeated now what we're going to do next is we're going to do some

questions where we will come across some restrictions okay so here's the first question let's see what it says so it

says find the number of ways in which all five letters in the word great can be arranged so notice that in the first

part there is no restrictions so so that means and there is no letter that's being repeated so that means the answer

to the first part will simply be 5 factorial which is equal to 120 now in the second part let's see what it says

how many of these arrangements are the letters how many of these arrangements are the letters a and a a and e next to

each other okay so what we're going to do is we're going to put a and e together we have to make sure that

they're next to each other now whether they are in the first and the second place or the second and the third place

or the third and the fourth place or the fourth and the fifth SP that does not matter as long as they are together okay

so how do we do questions like these now there is a standard way of putting objects together which is by the way one

type of question which we're going to do in detail we're going to do multiple possible questions of but for now for

starters here's what you guys need to understand so make five blanks since there are five letters in the word great

put the two letters that are supposed to be together which are A and E in a box okay when I say in a box that means you

basically since these are two letters you're going to obviously have one letter per blank but we're going to make

a small box around it so that we make sure that they can stay together okay now if you think about it A and E can

Shuffle amongst themselves I mean there's nothing in the question which says that a has to come first and then e

so that means I can Shuffle A and E and since these are two different letters I can shuffle them two factorial number of

a okay so this is what's happening inside the box but like I said since A and E have to be together and there's

nothing in the question which says where they they should be together so that means we can have AE anywhere okay so

how do we take that into account so now what we're going to do is we're going to treat AE as one unit okay so if I treat

AE as one unit how many units or how many different objects or or we're not going to call them letters okay but how

many different objects or units do I have to arrange so here's one unit which although is made up of two units but we

don't have to worry about that here's the second here's the third here's the fourth so that means we have four

different units which we have to arrange and how do you arrange four objects or four people or four letters or four

units in a straight line you do that by simply doing four factorial so that means you have 2 factorial multiplied by

4 factorial okay so this 2 factorial I repeat is representing what's going on inside that is basically considering AE

or EA and this four factorial is representing what's going on outside okay that and outside if you look at it

you have four different objects which you have to arrange and you can do that four factorial number of w okay now

let's see what the answer to that is the answer to that is well let's work it out 4 factorial *

2 which is equal to 48 so that means you can arrange of these 120 48 arrangements are such where A and E are next to each

other okay so there you go now before I move on to the next part there's one one thing that I would like you to

understand that is I'd like you to understand what is the logic behind this 2 factorial into 4 factorial

so basically here's what we're doing let's duplicate this let's bring it here okay now if you think about it

let's let's not use the four factorial approach instead there's another thing there's another approach which I would

like to like you to understand and then that will help you understand this 2 factorial into four factorial approach

so as far as AE is concerned we can shuffle them to factorial number of ways okay now for a minute if I only focus on

these letters okay so here we can see that we have three different letters here okay

and we can arrange them how many ways we can arrange them three factorial number of a but like I said there's nothing in

the question which says that A and E have to be in the first place okay they can be in the second third fourth

wherever basically so that means it's possible another Arrangement while satisfying the Restriction would be and

do keep in mind by the way I should have mentioned this in the beginning that whenever you have a certain restriction

make sure that you always fulfill that restriction first okay so whenever there is a certain restriction make sure that

you fulfill that restriction first okay so another possibility would be that I first have any other letter other than a

and e and then I have AE okay so that means now we can see two different ways in which a e can be together another

possibility would be let's duplicate this that we have first we have two blanks okay where we can have two

letters rather than a and e and then a e and then the final letter another possibility would be if I duplicate this

that first we have three different letters okay and then we have AE now if you think about it how many different

ways do we have AE together we have four different ways so that means if we're using this approach that of 2 factorial

into 3 factorial we have to multiply this by four because there are four different ways or four different places

where A and E can be while still being together so if you think about it what's 3 factorial into 4 that's basically 3

into 2 into 1 into 4 which is as good as 4 into 3 into 2 into 1 so that is why we do 2 factorial into 4 factorial that's

why we did 2 factorial into 4 factorial earlier notice that it leads to the same answer that is 48 ways but I don't

recommend I obviously don't recommend this approach okay not because it's wrong because it's lengthy this is the

approach that I recommend and this is the approach that you will find in most of the books okay so I hope that's clear

now let's do another example and here it is so let's see what it says it says how many

different how many Arrangements of the word achieve are there if there are no restrictions okay so the first part

usually not all the time but usually it has no restrictions okay so the only thing that we have to worry about is if

there is a letter that's being repeated so altogether there are seven letters and we can see that the letter e is

perhaps the only letter that's being repeated so we divide by two factorial if you do this right you get

25520 as your final answer okay now let's talk about part two in part two notice that there is the question says

that the first letter is in a so what the question here has done is it has fixed the position of one letter so we

make seven blanks 1 2 3 4 5 6 7 yeah that's it 1 2 3 4 5 6 7 okay now an important note which I should have

written earlier and that is note okay that we shouldn't have two colons so the note is that

whenever whenever there is a restriction Whenever there is a restriction make

sure to first settle that okay or settle slash satisfy

that so whenever you have a particular requirement make sure that you first fulfill that before you do anything else

okay so like here for example the first letter is an A so I got to make sure that the first letter is an a now as far

as the remaining six are concerned notice that we have six remaining letters okay but do keep in mind that of

the six we still have the two e somewhere in these six letters so we have to divide by 2 factorial now here

since we just have a fixed we don't really have a lot of options other than just one so you can do 1 into 6

factorial / 2 factorial okay or you can might as well completely ignore this one and you would get basically

360 different ways and that's the answer to the second part now let's see what the third part says the third part says

that the letters a and I are to be together okay so how can we do that how can we keep a and I to be together so

very similar to the question that we just solved 1 2 3 4 5 6 7 so let's place a here let's place I here now these two

are to be together which means that they can Shuffle amongst themselves there's nothing in the question which says that

a and I are to be together in that order there's no nothing of that sort although in some questions there will be so watch

out for that so that means we have 2 factorial which is considering the number of ways these letters can Shuffle

amongst themselves now what are the total number of units that we have to arrange now that's 1 2 3 4 5 6 that

means 2 factorial multiplied 6 factorial which is equal to what well wait a minute Not Just 2 factorial into 6

factorial don't forget that amongst the remaining six letters there are two e that we are repeating so in order to uh

in order to make sure that we don't end up up counting them as multiple Arrangements like we did with the

example Bob we had divide by 2 factorial so what do we get we get 720 ways okay now let's do part four

part four says the letter C and H are to be a part okay now notice there are two objects two letters that we have to make

sure are not together okay now what we're doing right now is something that applies to two objects that is the

number of ways that they are a part is equal to the total minus the number of ways that they are together okay so as

far as two objects are concerned they're either going to be together or they're not together okay there's nothing in

between so and since in the previous part where we figured the letters in and I are to be together and in this part

we're figuring out the letter C and H are to be apart so that means if I want to find out the number of way that C and

H are to be a part I need to First find out the number of a that they are together now if you think about it let's

do the working for that let's find out the number of is that they are together if you've probably if you've already

thought of what I am going to say then that's great 1 2 3 4 5 6 7 okay now the working for keeping C and H together

like I said if you think about it is not going to be any different from the working for a and I to be together why

because two letters although different but the number of letters are the same so you do two factorial for C and H and

for the remaining you 6 factorial divide by 2 factorial since we have to um divide by the number of letters that are

repeating and we would get the exact 720 okay now 720 is the number of ways that they are going to be together and What's

the total the question has already made us work out the total which is 2520 so 2520 the number of ways that

they are uh that they can be arranged minus the number of ways that they are together and that gives us 1,800 ways so

that means there are all together 1,800 ways in which cnh ought to be a part so what I was going to say was that you

don't really have to do this working because we've already done it for two other letters and it's going to be the

same so all you got to do is together a total minus the number of way that they are together which is equal to the

number of ways that they are going to be apart which is 1,800 okay now let's do the last example

this example says three girls and two boys are to be seated in a row find the number of different ways that this can

be done if the girls and boys sit alternately okay so three girls and two boys so that means we have all together

five people so let's make five blanks 1 2 3 4 5 okay now what should I start with what if I start with a boy and then

girls considering that they have to sit alternately and then a boy and then a girl and then a boy is this possible

let's see well this means that we have to have three boys and two girls well as far as two girls are concerned we can do

that since we have three girls but where will I get the third boy from so that means this is not possible so how do we

make it possible well we start with a girl so 1 2 3 4 5 so if I have a girl in the first place then I got to have a boy

in the second and a girl in the third and then a boy in the fourth and then a girl in the fifth okay so if you think

about it for the first place we have three options as far as a girl is concerned and for the next girl we have

two options since we have three girls all together and then for the final girl we have one option and then as far as

the boys are concerned we have two option for the first one two options for the first one and one for the last one

the second one so Al together if you think about it it's basically three factorial for the girls multiplied by 2

factorial you don't necessarily have to write factorial you can do 3 into 2 into 2 into 1 into one the way that you see

it and either way the answer is going to be 12 different ways okay so here also you can see that this is a certain type

of question where position of one or even multiple objects is fixed okay so that's the answer to part A Part B says

a girl sit at each end of the row so once again you can see the condition is that position of two

people are already fixed by the question so you got to have a girl in the actually let's write it down you got to

have a girl in the first place and a girl in the last space so how many options do you have since we have to

settle the Restriction first so for the first place we have three options since there are three girls to choose from and

in the last place we have two options why because we have two girls to choose from now what about in between in

between anything can happen and we don't really care because we have met the restriction or we have satisfied the

restrict the condition that was placed in the question so we are now left with how many people let's see so we had

three boys and two girls to start with so that was five now from five two girls have already been arranged that means

now we're down to three so we have to arrange three people from three people okay so we have we have three people to

arrange and we have to arrange all three of them so that can be done 3 factorial or 3 P3 number of ways so that means 3

into 3 factorial into 2 which is equal to 36 ways no yeah 36 ways 36 ways all right now part

C part C says the girl sit together and the boy sit together all right so once again let's make five blanks 1 2 3 4 5

so the girls have to sit together so that means here we have have the girl sitting together girl girl girl okay now

you might think that these are three girls and uh so why why aren't we dividing by three factorial well since

they're three different people we're not going to consider them as repeated objects okay so that is something that's

done when we're using numbers since if let's say you have two twos so one two is the same as the other two or if let's

say you have two P's one p is the same as the other P but when you when you have people even if let's say you have

three girls or four girls or five boys we're not going to say that oh because the boys are being repeated let's divide

it by the number of times they're repeating and the factorial of the number of times that they're repeating

okay every person is different from another basically that's the reason why so in this box we can have three girls

okay and remember that they can Shuffle amongst themselves so how can they Shuffle amongst themselves three

factorial okay or if you want to see the working behind it so here we have girl one girl two girl three and since they

can Shuffle amongst themselves that would be 3 factorial here we will place the boy since the question says that the

girls and boys are to be together so boy one and boy two and these boys there's obviously there's no restriction as to

why they can't Shuffle amongst themselves as long as they're together that's all that matters also another

thing that we have to consider is that the question doesn't say that the girls and boys the girls sit together and the

boys sit together in that order that means you can have all the boys sitting together first and then all the girls so

that means we now have to consider them as two different units and find out how many ways these two units can be

arranged which will basically be 2 factorial okay so 3 factorial * 2 factorial * 2 factorial is equal to what

so that's four I don't encourage that you work this out mentally it's best that you use your calculator 3 factorial

* 2 * 2 so that's equal to 24 different ways all right so yeah that's it that's it for this video I hope you guys have

understood and I hope you guys have understood the basic idea behind permutations and combinations in the

next video what we we're going to do is we're going to pick a specific type which is type one where the position of

one or more object is fixed and we're going to focus only on those type of questions and we're going to be doing

questions from P papers the questions that I solved in this video were not from ppers they were from different

books just so that you can sort of introduce uh get yourself introduced to the idea and I hope it's clear now so

that's it for this video I'll see you guys in the next one until then take care bye-bye

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