Hello everyone, welcome back to a new video In today's
video, we are going to do a topic that is Permutation and
Combination the topic which makes kids worried they are
afraid of this topic but in today's class, we will learn
how to do it with basic examples
then we will see their formula and definition and what is
the difference between these two how is permutation
different from combination and how is combination different
from permutation and we will solve questions on these so
let's start permutation
and combination first of all, I will write some specific
words for these suppose, where is permutation and
combination used if we see widely then it is used in
selection when we have to select some things from a bunch
of objects like we have 10 balls
we have to select 5 balls from it so in this case, what
happens is it becomes a case of selection suppose, out of
something, total was 10 balls number of total objects
became 10 and we have to select 5 balls so how many objects
we have to select 5 out
of 10 objects so we are selecting these objects so in this
case, in this example we have permutation and combination
used which one will be used, permutation or combination
that is later but now, how will we know
what type of examples we use we use in selection examples
so when it comes to selection so whatever you are selecting
that becomes a case of combination selection is a case of
combination suppose, you have n objects n means total
number of objects total
objects objects means any object like we take a ball any
element, whatever we are talking in question we call it an
object in normal math language so we have n, total number
of objects how many objects we have to select suppose, we
have to select r that
is r objects we have to choose how many objects we have to
choose we have to choose r objects or we have to select so
what is used in this we call it combination when we are
selecting but when along with selection we have to arrange
it suppose, we have
10
I just said 10 balls we have to select 5 balls but now
those 5 balls can come in any random order first it can
come of any color then it can come of any color so that
case in which your order will not matter that which ball or
which object we are choosing
in which arrangement will not matter that case will be of
combination in which you have to focus on selection later
on, arrangement or in which order which thing came first or
which thing came on second number that will not matter that
will be your combination
and obviously opposite in which we are selecting things
along with selection we are also arranging things so that
case will be of permutation for example, the smallest
example from which you will understand the difference that
is, let suppose like in
the example of ball select any ball first or later it does
not matter that in which order you have selected but if we
have a case of pin that you have to make a word or pin
using few objects so in that your selection will matter
plus in that your arrangement
will also matter suppose you have two alphabets
A and B you will choose them so their selection can be of
only one way that is, we have selected A also so this is
our one way so this is the example of combination that we
have selected only but if we have to generate a pin with
the help of this of two
digits using A and B so in that how many things we will get
that is, A will also be a different arrangement and B will
also be a different arrangement this will be a different
pin and this will be a different pin so in both the cases,
the objects are of same use that is A and B but as soon as
we change their order
change their placement so we are getting two ways of pin
whether we change the order in selection or not we will get
the same matter that will be the example of combination but
in which the order will matter the arrangement will matter
that will be the case of permutation
I hope it will be clear to you that if we have to choose
two alphabets then we can do it in one way or we can do it
in both ways
but if we have to arrange it along with choosing then we
will get two different arrangements one will be A and the
other will be B so permutation will always be more than
combination so the first difference is that in combination
we only select the objects
and in permutation we select the objects plus we also
arrange the objects this is your first difference now the
second difference will be the difference in their formula
that is what will be the formula of both now permutation
means if we are asked that
you have total n objects how many you have to choose out of
that choose r objects so simply they are choosing and we
are not arranging the rest so we will use the formula of
combination the formula of combination will be n, c, r n, c
for combination and
r in the subscript what does it mean? choosing r objects
choosing or selecting r objects from n number of objects
means you have total n number of objects suppose you have
10 you have to choose 5 out of them so what will happen?
10, c, 5 means total 5 balls you have 10 balls you have to
choose only 5 so 10,
c, 5 would be the formula for that how will we open this?
I will tell you now first we will do the permutation in
permutation what will be your formula? here n, c, r was
being used here n, p, r will be used p for permutation n,
p, r means we are selecting r objects from n number of
objects select plus arrange
also we are selecting but along with that we are also
arranging we are changing their order and making different
new terms like if we want to make a word we have 5 letters
Delhi Delhi has 5 letters now if we are using 5 letters so
if we choose 5 letters and put them randomly then it will
be a combination but if we arrange them like Delhi will be
this combination second can be d, e, l i and h can be
exchanged so we have
a new word but this is not Delhi this is a different word
now if I replace these two so it will be e, d, l, i, h
Even if you have to change the word of the previous
sentence, the word of the previous sentence will be the
same.
But if you have to change the word of the previous
sentence, the word of the previous sentence will be a
different word.
The arrangement of the sentence is important.
Now, what is the NPR and NCR? we have given the formula of
permutation as NPR and the formula of combination as NCR
but what will we solve them as NCR so to solve them first
of all i would like to discuss one small thing with you
that is factorial you
might not have heard of it or if you have heard of it then
factorial what is factorial of any number we will find out
factorial of any number and we will always find out of
natural number we will also find out of zero but we will
see what happens but
we will find out of natural number so suppose you have to
find out 5 factorial sign is like this
so 5 factorial how will we read 5 factorial now how will we
find out its value whatever number you have written 5, 5
whichever you have to find out first write that then
multiply one small natural number with that what is small
natural number from 5,
4 so one will go back that is 4 multiply it then one will
go back that is 3 then till when we will go back till 1
till we don't get 1 in backwards we will multiply till
there so what is 5 factorial 5 x 4 x 3 x 2 x 1 whatever we
will get after solving
that will be 5 factorial solve it 120 will come 5 x 4 is 20
and 20 x 6 is 120 so 120 will be 5 factorial either you
understand it like this or you understand it like this that
we will start from 1 natural number 1 then 2 and we will go
up one by one 1,
2, 3, 4, 5 and till where we will go till the number of
factorial we have been asked suppose you have been asked 10
factorial so from where you will start 10 factorial we will
start from 1 1 x 2 x 3 x till where we will do x till 10 x
9 x 10 now after
multiplying this whatever answer will come that will be
your 10 factorial if you understand the meaning of
factorial then I will write the simple definition of
factorial if we have to find n factorial where n is any
natural number then what we will do
we will start multiplication from 1 1, 2, 3 and so on then
we will do last you should get n last x n we will write the
number before n before n what will happen n-1 n-1 like this
we will go back and back till 1 so all this multiplication
whatever will
come that will be your n factorial so like this we will see
the formula of npr npr will be equal to n factorial now
either you denote factorial like this or you denote like
this it is same or you write like this both are same n
factorial so n factorial
whatever is the total number of objects n factorial upon
n-r factorial do the difference of both and take out its
factorial so npr will be equal to n factorial divided by
n-r factorial now same ncr will be equal to you will have
ncr first you write same
as npr n factorial upon n-r factorial same npr just one
thing will be added in denominator we will add into r
factorial
in denominator we will add into r factorial that's all you
will get the formula formula of ncr so ncr i said all this
is p npr so what i can write this 1 upon r factorial into
instead of this i will write npr npr i have written
now i can multiply this r factorial so you will get the
relation r factorial into ncr is equal to npr this is the
relation between combination formula and the permutation
formula npr is equal to r factorial into ncr this means we
have n total objects
and how many r we have to choose or select that understand
the relation now write properties after writing this
relation properties what will be the properties of
factorial not of factorial but of these formulas you
understood factorial
now write 0 factorial 0 factorial is equal to 1 0 factorial
is equal to 1 and 1 factorial is also equal to 1 0
factorial is also equal to 1 and 1 factorial is also equal
to 1 now next ncr if you take out ncr or nc n minus r both
will be same because in
both the denominator same thing will come now if i open
with formula so this formula you know what will be ncr n
factorial divide by n minus r factorial into r factorial
this is ncr now if i take out formula of this side so this
will be n factorial upon
n minus r this one its factorial into one more factorial is
from n this minus this so n minus n minus r factorial
understood this is that term n minus r term now we have n
minus r so we have written n minus n minus r factorial ok
now multiply this minus
sign inside this n will cancel what will be left plus r so
this formula we have n factorial divided by n minus r
factorial into r factorial see this is equal to this
formula which we have written here n minus r factorial so
these two terms are ncr is
equal to nc n minus r for example if we have value of n as
10 total we have 10 take value of r as 2 so nc2 sorry put
value of n 10cr 10cr is 2 10cr will be equal to 10c2 will
be equal to 10c now what you do minus 2 from 10 that is 8
so 10c8 10c2 and 10c8
what will be the value of these two value of these two will
be equal will be same ok by this formula you have already
read the relation you note this down once
I will read it next you have written this formula next now
what we have there are two types of principles to solve the
questions of permutation and combination
wait before telling that let us do one question wait let us
do one question this is your question that is if 6Pr is
equal to 360
I hope it's visible. You're right. If 6Pr is equal to 360
and if 6Cr is equal to 15, then find the value of r.
Then, we are given the value of 6Pr and 6Cr and we have to
find out the value of r.
So, use their relation. What was their relation? It was r!
into nCr is equal to nPr.
Now, if we use r! into n. What is the value of r? We have 6
instead of n. So, put 6 instead of n.
6Cr is equal to 6Pr. Put the value. 6Cr is equal to 15 and
6Pr is equal to 360.
So, r! is equal to 360 divided by 15. Cancel it out.
So, we have the value of r which is 24.
So, you have the value of r! which is 24. So, you have to
find out the number which has the answer of 24.
Now, when we read the factorial, we found out the value of
5! which was 120.
So, it can't be 5! nor it can be greater than 5! because
the answer will increase by multiplying.
So, let's see the answer of 4! which is less than 5!
The answer of 4! will be 4 into 3 into 2 into 1. That is 4
into 3 is 12. 12 into 2 is 24. 24 into 1 is 24.
So, the value of 24 is 4! Now, r! is equal to 4!
So, we have the value of r!
Easy! This was an easy question.
Now, because we have done this property, so I have done
this practice question.
Next, what I wanted to tell you was two main principles or
fundamental principles in permutation and combination.
That is the fundamental principle of multiplication and the
other is addition.
Addition and multiplication. So, the principle of
multiplication, I will simply write this.
When will you use multiplication and when will you use
addition?
Now, what is multiplication and addition?
When will we multiply the answers we have and when will we
add those answers to each other?
How will we know this difference?
So, I will give a simple example.
Whenever you have, note one more thing, whenever you have
the case of AND,
AND means you have two conditions and those two things or
condition or event, whatever you name it, when those two
things simultaneously, means at the same time, should be
selected or should be chosen, then we use the principle of
multiplication.
And when we have the sense of OR, like you have a choice,
that either you do this or you do that, then either you
select this or you select that.
So, when this case of OR comes, either this or this, then
which principle will we use?
We will use the principle of addition.
Then, what will we do to the things? We will add them.
I have just verbally told you that in AND, there is
multiplication and in OR, there is addition.
But, how will we do it in the question?
So, you see an example.
Let's suppose you have twenty, twenty children's class,
that is, how many total students are there? Twenty in a
class, total students.
Now, what you have been told is that you have to select two
students, select, select, any two students, two students,
we have to select out of these twenty students.
Now, out of those two children, out of these twenty
students, out of these twenty, you have twelve boys and
eight girls. The children in your class, twenty in total,
in which order are they divided in genders? Twelve are boys
and eight are girls.
Now, you have to select two students as a class
representative.
Now, those two students can be anyone.
So, if you are not told anything, just you have to select,
you are not told that it should be a girl or a boy, then
you are doing a simple selection. And first select someone,
then select someone, order will not matter in selection.
So, this will be the case of combination.
So, what you have to do is, out of twenty, how many have to
be selected? Two have to be selected. So, 20C2 would be the
formula for that.
That is, selecting two students among twenty students.
Out of twenty students, what are we doing to two students?
We are selecting.
So, it is a simple combination formula.
Because it is a selection.
But, if we are asked this question, that the two students
we are selecting, out of these two students, one should be
a boy and one should be a girl. In this way, you have to
choose a representative.
That is, out of the two students you have selected, one
should be a boy and the other should be a girl.
So, how will you select this?
So, first of all, your first condition is, what should be
one of them?
It should be a boy.
So, the first condition is, it should be a boy.
So, from which section will we select the boy?
We will select from the section of boys.
So, in total we have twelve choices.
How many boys are there in total? There are twelve boys.
And how many of the twelve should we select?
We have to select one boy.
So, this will be selecting one boy.
You will have the possibility to select one boy.
12C1 Means selecting one boy.
Not one boy, one boy.
We are selecting one boy.
Out of twelve boys.
So, this is the number of ways to select one boy.
Now, what we have to do is, we have to choose a girl.
We have to choose a girl.
So, we will select the girl from the girl group.
What will that be? 8C1 We have the possibility to select
one girl.
What will that be? 8C1 Out of the total eight girls, we
have selected one girl.
Now, we have two events.
One is the event of selecting a boy.
The other is the event of selecting a girl.
Now, are these two events happening together or separately?
Now, we have to do both together.
Because, in total we need two children.
So, when these two events are happening together, then what
will these two things be?
These two things will multiply each other.
What will they be?
They will multiply each other.
Because, this is only one event.
These two events have to happen together.
They are not happening separately.
This is not our case.
That you select one boy or one girl.
You have to select one boy and one girl.
So, as soon as the case of AND comes, we will use the
principle of multiplication.
So, what will these two things be?
They will multiply each other.
Now, if you are not told that one boy and one girl should
be there.
You are told that either it is a team of two boys, which
you are selecting, or it is a team of two girls.
Okay? The team which you are selecting, there will be
either two boys, or you have to choose the same gender, or
there will be two girls.
So, how will you question this?
Now, as soon as the AND comes, either it is this case or
that case.
So, there should be one possibility out of both.
So, what we will do is, we will select both individually,
and we will add that possibility to each other.
So, let's add.
select 2 boys total boys are 12 select 2 boys selecting 2
boys among 12 boys next is select 2 girls among 8 girls
total 8 girls are 2 now these are 2 different events these
2 events are not related to each other either this event
will happen or that event
will happen so what will be our principle in this principle
of addition now these 2 things will be added so i hope you
have understood where to multiply and where to add this was
the intro of permutation and combination in next video
because this video
will be long and you may not understand everything so in
next video we will do questions after this video please
practice this don't practice this, think about real world
problem in which you are using permutation and in which you
are using addition and
think about which example is used for permutation and which
is used only for combination this is your homework to find
out real life example from your side so see you in next
video till then you practice thank you for watching the
video
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