Class 11 Exam > Class 11 Videos > L11 : Examples-1- Binomial theorem, Mathematics, Class 11
L11 : Examples-1- Binomial theorem, Mathematics, Class 11 Video Lecture
FAQs on L11 : Examples-1- Binomial theorem, Mathematics, Class 11 Video Lecture
| 1. What is the binomial theorem? |  |
Ans. The binomial theorem is a mathematical formula that provides a way to expand the powers of a binomial expression. It states that for any positive integer n, the expansion of (a + b)^n can be given by the sum of terms of the form C(n, r) * a^(n-r) * b^r, where C(n, r) represents the binomial coefficient and is calculated using the formula n! / (r! * (n-r)!).
| 2. How is the binomial theorem used in mathematics? | |
Ans. The binomial theorem is used in various branches of mathematics, such as algebra, combinatorics, and calculus. It allows us to quickly expand the powers of a binomial expression, making calculations easier. It is particularly useful in solving problems involving binomial coefficients, probabilities, and generating functions.
| 3. Can you provide an example of using the binomial theorem? | |
Ans. Sure! Let's say we want to expand (x + y)^3 using the binomial theorem. According to the theorem, the expansion will be:
(x + y)^3 = C(3, 0) * x^3 * y^0 + C(3, 1) * x^2 * y^1 + C(3, 2) * x^1 * y^2 + C(3, 3) * x^0 * y^3
Simplifying this expression, we get:
(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3
So, the expansion of (x + y)^3 using the binomial theorem is x^3 + 3x^2y + 3xy^2 + y^3.
| 4. What is the significance of binomial coefficients in the binomial theorem? | |
Ans. Binomial coefficients play a crucial role in the binomial theorem. They determine the coefficients of each term in the expansion of a binomial expression. Binomial coefficients can be calculated using the formula C(n, r) = n! / (r! * (n-r)!), where n is the power of the binomial and r is the index of each term. These coefficients provide information about the number of ways to choose r objects from a set of n objects, which has applications in combinatorics and probability.
| 5. Can the binomial theorem be applied to expressions with more than two terms? | |
Ans. No, the binomial theorem is specifically designed for binomial expressions, which have only two terms. It cannot be directly applied to expressions with more than two terms. However, certain extensions and modifications of the binomial theorem, such as the multinomial theorem, allow for the expansion of expressions with more than two terms. These extensions involve additional coefficients and terms to accommodate the increased complexity of the expression.
Text Transcript from Video
hello friends this video binomial
theorem part 11 is brought to you by
exam fear calm no more fear from exam
before watching this video you should
have washed mine mu theorem part 1 -
part 10 so here is a problem where it
says that 2nd 3rd and 4th term of this
binomial expression this are this this
is this - 47 21 0 so we have to find
value of XA n so it says T 2 is equal to
240 T 3 is equal to 720 and E 4 is equal
to 1 z is not T 2 is a key the
coefficient of no the holdup the whole
term is going to this so it says T 2 is
equal to 2 for T 3 is equal to this
entry quiz herbalism so we can also say
T 2 we can say write T 2 is equal to get
it t 1 + 1 so we know a formula T R plus
1 is equal to NC R then and then we say
a to the power R an X to the power n
minus R so here T 1 plus 1 is equal to n
is what n n C R is 1 is a RS 1 into R 1
into H to the power n minus 1 and this
guy is equal to 1 this is first equation
similarly we can say here T 2 plus 1 is
equal to NC 2 a to the power 2 X to the
power n minus 2 is equal to 7 this is my
second equation also I can say T 4 I
guess 83 plus 1 and this is equal to n
3aq X to the power n minus 3 is equal to
1 0 so what I have done I have formed
three equations P 2 is going to 240 so I
can say t2 is equal to t1 plus 1 R plus
1 so duplicity the formula I got NC 1 a
HT 1 n minus 1 to 40 NC 2 a square X to
power n minus 2 is equal to 720 m NC 3 8
u X to the power n minus 3 is written 1
0 is it now let's divide these two
equations these two equations divide
this by this so I get 720 by 2 for the
because from here we have to find a xn n
n X and a are three variables we have
got three equations we have to solve it
not simple so we divide this by this
this by this is equal to actually mc2 a
square X to the power n minus 2 divided
by 10 see one a H to the power n minus 1
correct so if you divide this or 240 by
720 plus 6 is equal to hence you two we
can say n permutation by n minus 2
permutation
factorial and n C 1 we can say n
factorial n minus 1 pictorial perfect
real and a square by a is equal to a and
X to the power n minus 2 by n minus 1 is
going to X so this is what we got solve
it further n squares n squares and
contrarian module 1 n minus 2 factorial
and this is n minus is factly
so we can also say 6 value n minus 1
factor like and say n minus 1 into n
minus 2 factorial we can say and this is
n minus 2 13 and 2 factorial I can say 2
into 1 and this is a by X n minus 2
factorial I can cancel
or what I get six by X is equal to K
into n minus one this is my equation for
six by X equal to a 2-1 now similarly
what we can do we can divide these two
these two equations so if you divide
these two equations let me divide this 2
so what our leaders 1 0 8 0 / 7 20
this guy is equal to n c3 e to the power
3 X to the power n minus 3 divided by NC
2 a to the word 2 X to the power 3 minus
2 this is equal to n factorial n minus 3
factorial + 3 factorial and this is
equal to n c2 is equal to n factorial by
n minus 2 factorial and here 2 factorial
+ H cube by a square is going to 8 and
this becomes X 8 is become X or you
cancel this divide this by shakes
six six thirty six 6 12 72 and 606 48
cases of of the a if you divide further
we get 6 you get 9 this becomes and this
becomes 2 this becomes 3 3 by 2 and so
also we can say 3 by 2 is equal to n
factorial inverter we can cancel and n
minus 2 factorial I can say is equal to
n minus 2 into n
three back here and two-factor where I
guess a doing two one smell DN minus
three factorial I can write as it is
because this is n minus K factorial and
three better legacy painted into one so
I can cancel evidence to bacterial and
to when to also enhancer I get n minus
two but three into a when heroes events
or what I get as final equation this
comes here so what I get is a by X is
equal to do this I'm taking here three
into three goes up nine by E into this
is there so this also comes here by n
minus this is my fifth equation a by X
is metal 9 by 2 by n minus three now if
we divide this fight with the fourth
equation the fourth equation we have God
was a y is equal to 6 by n minus 1
because the equation we have got so if
we equate this to what we get is this
for the fourth equation we have got in
the last foil this is the fifth edition
equate this egg by six what I get 9 by 2
into n minus 2 is equal to since by n
minus 1 so now if we multiply this we
get 9 n minus 9 is equal to 12 n minus
sixty twelve twelve into 224 or we get
three n is equal to 24 minus nine will
be 50 or n is equal to PI so we have got
n is equal to 5 now this is an answer
now you to find the value of a n X so in
the first
equation we have got what we have got
240 is equal to X to the power 4 into if
so thick machines this is what we got
from the first equation and from the
fourth equation this equation then you
put n is equal to 5 what you get is 6 is
equal to K by X - 4 so now if you solve
both these these two equation e by XD 4
and this this 2 equation so what I get
is X is going to do and a is equal to
because we have two equations we have
two efficient this efficient we got from
here n is equal to 4 we put so we get a
by X is equal to 6 by 4 FX / - 6 by 4
and here we also we have for this one 2
X power 4 into 8 so here I can say is
equal to 3 by 2 X so we put this we get
or we can I can show you like this so
240 is equal to X to the power 4 into 8
a I can say is 3 by 2 X by 2 X so we
will multiply this divide this we get
into 40 into 2 by 3 is equal to nation
power 5
we still have 8 8 D so 160 is equal to
base to the power 5 sorry it was a
mistake here so in the first equation we
have got 240 is where 2 n X to the power
4 where n is good but 5 H to the power 4
because here in the first equation you
see write the prosecution is equal to
240 is equal to K 2 is equal to n C 1
into H to the power n minus 1 into a to
the power 1 so any to see what is going
to is equal to the file XE power n minus
1
four and a so here you have five also so
this is by five so this is 16 so X to
the power 5 is equal to 32 so X is equal
to 2x is going to do so if you put X
equal to 2 in this equation also 6 is
equal to a by 2 or a is equal to what
you get to L where for this is 1 2 3
because we have a is equal to 3 X is
equal to 2 and n is going to look like
this is the answer let's see what what
we have done once again so we have T 2 T
3 T 4 is going to 47-21 0 8 0 so we have
expanded this to your wave form of T 2 T
2 is equal to NC 1 e xt / n minus 1 t 3
is good 2 NC 2 a square X n minus 2 and
T 4 is going to n C 3 a cube X minus 3
and then we have divided these two
equation to get one equation and divided
equation 2 by 3 to get one equation so
we have got equation this it has five
and is an equation for now solving this
four and five we got n is equal to five
so we have now got one rather than is
equal to 5 so we have used equation one
and equation 4 to again solve and get
the value of n X because we already had
them below and is going to be PI thank
you visit exam clear calm to watch free
education videos try free online test it
the best quality study material study
from the best tutor inventors and much
forum times once again
theorem part 11 is brought to you by
exam fear calm no more fear from exam
before watching this video you should
have washed mine mu theorem part 1 -
part 10 so here is a problem where it
says that 2nd 3rd and 4th term of this
binomial expression this are this this
is this - 47 21 0 so we have to find
value of XA n so it says T 2 is equal to
240 T 3 is equal to 720 and E 4 is equal
to 1 z is not T 2 is a key the
coefficient of no the holdup the whole
term is going to this so it says T 2 is
equal to 2 for T 3 is equal to this
entry quiz herbalism so we can also say
T 2 we can say write T 2 is equal to get
it t 1 + 1 so we know a formula T R plus
1 is equal to NC R then and then we say
a to the power R an X to the power n
minus R so here T 1 plus 1 is equal to n
is what n n C R is 1 is a RS 1 into R 1
into H to the power n minus 1 and this
guy is equal to 1 this is first equation
similarly we can say here T 2 plus 1 is
equal to NC 2 a to the power 2 X to the
power n minus 2 is equal to 7 this is my
second equation also I can say T 4 I
guess 83 plus 1 and this is equal to n
3aq X to the power n minus 3 is equal to
1 0 so what I have done I have formed
three equations P 2 is going to 240 so I
can say t2 is equal to t1 plus 1 R plus
1 so duplicity the formula I got NC 1 a
HT 1 n minus 1 to 40 NC 2 a square X to
power n minus 2 is equal to 720 m NC 3 8
u X to the power n minus 3 is written 1
0 is it now let's divide these two
equations these two equations divide
this by this so I get 720 by 2 for the
because from here we have to find a xn n
n X and a are three variables we have
got three equations we have to solve it
not simple so we divide this by this
this by this is equal to actually mc2 a
square X to the power n minus 2 divided
by 10 see one a H to the power n minus 1
correct so if you divide this or 240 by
720 plus 6 is equal to hence you two we
can say n permutation by n minus 2
permutation
factorial and n C 1 we can say n
factorial n minus 1 pictorial perfect
real and a square by a is equal to a and
X to the power n minus 2 by n minus 1 is
going to X so this is what we got solve
it further n squares n squares and
contrarian module 1 n minus 2 factorial
and this is n minus is factly
so we can also say 6 value n minus 1
factor like and say n minus 1 into n
minus 2 factorial we can say and this is
n minus 2 13 and 2 factorial I can say 2
into 1 and this is a by X n minus 2
factorial I can cancel
or what I get six by X is equal to K
into n minus one this is my equation for
six by X equal to a 2-1 now similarly
what we can do we can divide these two
these two equations so if you divide
these two equations let me divide this 2
so what our leaders 1 0 8 0 / 7 20
this guy is equal to n c3 e to the power
3 X to the power n minus 3 divided by NC
2 a to the word 2 X to the power 3 minus
2 this is equal to n factorial n minus 3
factorial + 3 factorial and this is
equal to n c2 is equal to n factorial by
n minus 2 factorial and here 2 factorial
+ H cube by a square is going to 8 and
this becomes X 8 is become X or you
cancel this divide this by shakes
six six thirty six 6 12 72 and 606 48
cases of of the a if you divide further
we get 6 you get 9 this becomes and this
becomes 2 this becomes 3 3 by 2 and so
also we can say 3 by 2 is equal to n
factorial inverter we can cancel and n
minus 2 factorial I can say is equal to
n minus 2 into n
three back here and two-factor where I
guess a doing two one smell DN minus
three factorial I can write as it is
because this is n minus K factorial and
three better legacy painted into one so
I can cancel evidence to bacterial and
to when to also enhancer I get n minus
two but three into a when heroes events
or what I get as final equation this
comes here so what I get is a by X is
equal to do this I'm taking here three
into three goes up nine by E into this
is there so this also comes here by n
minus this is my fifth equation a by X
is metal 9 by 2 by n minus three now if
we divide this fight with the fourth
equation the fourth equation we have God
was a y is equal to 6 by n minus 1
because the equation we have got so if
we equate this to what we get is this
for the fourth equation we have got in
the last foil this is the fifth edition
equate this egg by six what I get 9 by 2
into n minus 2 is equal to since by n
minus 1 so now if we multiply this we
get 9 n minus 9 is equal to 12 n minus
sixty twelve twelve into 224 or we get
three n is equal to 24 minus nine will
be 50 or n is equal to PI so we have got
n is equal to 5 now this is an answer
now you to find the value of a n X so in
the first
equation we have got what we have got
240 is equal to X to the power 4 into if
so thick machines this is what we got
from the first equation and from the
fourth equation this equation then you
put n is equal to 5 what you get is 6 is
equal to K by X - 4 so now if you solve
both these these two equation e by XD 4
and this this 2 equation so what I get
is X is going to do and a is equal to
because we have two equations we have
two efficient this efficient we got from
here n is equal to 4 we put so we get a
by X is equal to 6 by 4 FX / - 6 by 4
and here we also we have for this one 2
X power 4 into 8 so here I can say is
equal to 3 by 2 X so we put this we get
or we can I can show you like this so
240 is equal to X to the power 4 into 8
a I can say is 3 by 2 X by 2 X so we
will multiply this divide this we get
into 40 into 2 by 3 is equal to nation
power 5
we still have 8 8 D so 160 is equal to
base to the power 5 sorry it was a
mistake here so in the first equation we
have got 240 is where 2 n X to the power
4 where n is good but 5 H to the power 4
because here in the first equation you
see write the prosecution is equal to
240 is equal to K 2 is equal to n C 1
into H to the power n minus 1 into a to
the power 1 so any to see what is going
to is equal to the file XE power n minus
1
four and a so here you have five also so
this is by five so this is 16 so X to
the power 5 is equal to 32 so X is equal
to 2x is going to do so if you put X
equal to 2 in this equation also 6 is
equal to a by 2 or a is equal to what
you get to L where for this is 1 2 3
because we have a is equal to 3 X is
equal to 2 and n is going to look like
this is the answer let's see what what
we have done once again so we have T 2 T
3 T 4 is going to 47-21 0 8 0 so we have
expanded this to your wave form of T 2 T
2 is equal to NC 1 e xt / n minus 1 t 3
is good 2 NC 2 a square X n minus 2 and
T 4 is going to n C 3 a cube X minus 3
and then we have divided these two
equation to get one equation and divided
equation 2 by 3 to get one equation so
we have got equation this it has five
and is an equation for now solving this
four and five we got n is equal to five
so we have now got one rather than is
equal to 5 so we have used equation one
and equation 4 to again solve and get
the value of n X because we already had
them below and is going to be PI thank
you visit exam clear calm to watch free
education videos try free online test it
the best quality study material study
from the best tutor inventors and much
forum times once again
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