Detailed Notes: Binomial Theorem | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


Binomial Theorem 
Binomial expression : 
Any algebraic expression which contains two dissimilar terms is called binomial 
expression. 
For example : ?? + ?? , ?? 2
?? +
1
?? ?? 2
,3- ?? ,v?? 2
+ 1 +
1
(?? 3
+1)
1/3
 etc. 
Terminology used in binomial theorem: 
Factorial notation : n or n ! is pronounced as factorial n and is defined as 
?? ! = {
?? (?? - 1)(?? - 2)…….3 2.1 ;
1 if ?? ? ?? 1 if ?? = 0
 
Note : n! = n.(n- 1)! ; n ? N 
Mathematical meaning of  
?? ?? ?? : The term  
?? ?? ?? denotes number of combinations of ?? 
things choosen from ?? distinct things mathematically,  
?? ?? ?? =
?? !
(?? -?? )! ?? !
,?? ,?? ? ?? ,0 = ?? = ?? 
Note : Other symbols of of  
?? ?? ?? are (
?? ?? ) and ?? (?? ,?? ) . 
Properties related to  
?? ?? ?? : 
(i)  
?? ?? ?? = 
?? ?? ?? -?? 
Note: If  
?? ?? ?? = 
?? ?? ?? ? Either ?? = ?? or ?? + ?? = ?? 
(ii)   
?? ?? ?? + 
?? ?? ?? -1
= 
?? +1
?? ?? 
(iii) 
 
?? ?? ?? 
?? ?? ?? -1
=
?? -?? +1
?? 
(iv)   
?? ?? ?? =
?? ?? 
?? -1
?? ?? -1
=
?? (?? -1)
?? (?? -1)
 
?? -2
?? ?? -2
= =
n(n-1)(n-2)………(n-(r-1))
r(r-1)(r-2)…….2.1
 
(v) If n and r are relatively prime, then  
n
C
r
 is divisible by n. But converse is not 
necessarily true. 
Statement of binomial theorem : 
(?? + ?? )
?? = 
?? ?? 0
?? ?? ?? 0
+ 
?? ?? 1
?? ?? -1
?? 1
+ 
?? ?? 2
?? ?? -2
?? 2
+ ?+ 
?? ?? ?? ?? ?? -?? ?? ?? + ?…+ 
?? ?? ?? ?? 0
?? ?? 
where n ? N 
Page 2


Binomial Theorem 
Binomial expression : 
Any algebraic expression which contains two dissimilar terms is called binomial 
expression. 
For example : ?? + ?? , ?? 2
?? +
1
?? ?? 2
,3- ?? ,v?? 2
+ 1 +
1
(?? 3
+1)
1/3
 etc. 
Terminology used in binomial theorem: 
Factorial notation : n or n ! is pronounced as factorial n and is defined as 
?? ! = {
?? (?? - 1)(?? - 2)…….3 2.1 ;
1 if ?? ? ?? 1 if ?? = 0
 
Note : n! = n.(n- 1)! ; n ? N 
Mathematical meaning of  
?? ?? ?? : The term  
?? ?? ?? denotes number of combinations of ?? 
things choosen from ?? distinct things mathematically,  
?? ?? ?? =
?? !
(?? -?? )! ?? !
,?? ,?? ? ?? ,0 = ?? = ?? 
Note : Other symbols of of  
?? ?? ?? are (
?? ?? ) and ?? (?? ,?? ) . 
Properties related to  
?? ?? ?? : 
(i)  
?? ?? ?? = 
?? ?? ?? -?? 
Note: If  
?? ?? ?? = 
?? ?? ?? ? Either ?? = ?? or ?? + ?? = ?? 
(ii)   
?? ?? ?? + 
?? ?? ?? -1
= 
?? +1
?? ?? 
(iii) 
 
?? ?? ?? 
?? ?? ?? -1
=
?? -?? +1
?? 
(iv)   
?? ?? ?? =
?? ?? 
?? -1
?? ?? -1
=
?? (?? -1)
?? (?? -1)
 
?? -2
?? ?? -2
= =
n(n-1)(n-2)………(n-(r-1))
r(r-1)(r-2)…….2.1
 
(v) If n and r are relatively prime, then  
n
C
r
 is divisible by n. But converse is not 
necessarily true. 
Statement of binomial theorem : 
(?? + ?? )
?? = 
?? ?? 0
?? ?? ?? 0
+ 
?? ?? 1
?? ?? -1
?? 1
+ 
?? ?? 2
?? ?? -2
?? 2
+ ?+ 
?? ?? ?? ?? ?? -?? ?? ?? + ?…+ 
?? ?? ?? ?? 0
?? ?? 
where n ? N 
or  (?? + ?? )
?? = ?
?? =0
?? ? 
?? ?? ?? ?? ?? -?? ?? ?? 
Note : If we put ?? = 1 and ?? = ?? in the above binomial expansion, then 
or  (1+ ?? )
?? = 
?? ?? 0
+ 
?? ?? 1
?? + 
?? ?? 2
?? 2
+ ?+ 
?? ?? ?? ?? ?? + ?+ 
?? ?? ?? ?? ?? 
or  (1+ ?? )
?? = ?
?? =0
?? ? 
?? ?? ?? ?? ?? 
Problem 1 : Expand the following binomials : 
(i)  (?? + v2)
5
 
(ii)  (1-
3?? 2
2
)
4
 
Solution : 
(i)  (?? + v2)
5
= 
5
?? 0
?? 5
+ 
5
?? 1
?? 4
(v2)+ 
5
?? 2
?? 3
(v2)
2
+ 
5
?? 3
?? 2
(v2)
3
+ 
5
?? 4
?? (v2)
4
+
 
5
?? 5
(v2)
5
 
= ?? 5
+ 5v2?? 4
+ 20?? 3
+ 20v2?? 2
+ 20?? + 4v2 
(ii) 
(1 -
3?? 2
2
)
4
 = 
4
?? 0
+ 
4
?? 1
(-
3?? 2
2
) + 
4
?? 2
(-
3?? 2
2
)
2
+ 
4
?? 3
(-
3?? 2
2
)
3
+ 
4
?? 4
(-
3?? 2
2
)
4
 = 1- 6?? 2
27
2
+ ?? 4
-
27
2
?? 6
+
81
16
?? 8
 
Problem 2 : Expand the binomial (
2
?? + ?? )
10
 up to four terms 
Solution :  (
2
?? + ?? )
10
= 
10
?? 0
(
2
?? )
10
+ 
10
?? 1
(
2
?? )
9
?? + 
10
?? 2
(
2
?? )
8
?? 2
+ 
10
?? 3
(
2
?? )
7
?? 3
+ ? 
Observations : 
(i) The number of terms in the binomial expansion (?? + ?? )
?? is ?? + 1. 
(ii) The sum of the indices of ?? and ?? in each term is ?? . 
(iii) The binomial coefficients (  
?? ?? 0
, 
?? ?? 1
……….. 
?? ?? ?? ) of the terms equidistant from the 
beginning and the end are equal, i.e.  
?? ?? 0
= 
?? ?? ?? , 
?? ?? 1
= 
?? ?? ?? -1
 etc. {? 
?? ?? ?? = 
?? ?? ?? -1
} 
(iv) The binomial coefficient can be remembered with the help of the following pascal's 
Triangle (also known as Meru Prastra provided by Pingla) 
Index of the binomial 
The binomial coefficient 
Page 3


Binomial Theorem 
Binomial expression : 
Any algebraic expression which contains two dissimilar terms is called binomial 
expression. 
For example : ?? + ?? , ?? 2
?? +
1
?? ?? 2
,3- ?? ,v?? 2
+ 1 +
1
(?? 3
+1)
1/3
 etc. 
Terminology used in binomial theorem: 
Factorial notation : n or n ! is pronounced as factorial n and is defined as 
?? ! = {
?? (?? - 1)(?? - 2)…….3 2.1 ;
1 if ?? ? ?? 1 if ?? = 0
 
Note : n! = n.(n- 1)! ; n ? N 
Mathematical meaning of  
?? ?? ?? : The term  
?? ?? ?? denotes number of combinations of ?? 
things choosen from ?? distinct things mathematically,  
?? ?? ?? =
?? !
(?? -?? )! ?? !
,?? ,?? ? ?? ,0 = ?? = ?? 
Note : Other symbols of of  
?? ?? ?? are (
?? ?? ) and ?? (?? ,?? ) . 
Properties related to  
?? ?? ?? : 
(i)  
?? ?? ?? = 
?? ?? ?? -?? 
Note: If  
?? ?? ?? = 
?? ?? ?? ? Either ?? = ?? or ?? + ?? = ?? 
(ii)   
?? ?? ?? + 
?? ?? ?? -1
= 
?? +1
?? ?? 
(iii) 
 
?? ?? ?? 
?? ?? ?? -1
=
?? -?? +1
?? 
(iv)   
?? ?? ?? =
?? ?? 
?? -1
?? ?? -1
=
?? (?? -1)
?? (?? -1)
 
?? -2
?? ?? -2
= =
n(n-1)(n-2)………(n-(r-1))
r(r-1)(r-2)…….2.1
 
(v) If n and r are relatively prime, then  
n
C
r
 is divisible by n. But converse is not 
necessarily true. 
Statement of binomial theorem : 
(?? + ?? )
?? = 
?? ?? 0
?? ?? ?? 0
+ 
?? ?? 1
?? ?? -1
?? 1
+ 
?? ?? 2
?? ?? -2
?? 2
+ ?+ 
?? ?? ?? ?? ?? -?? ?? ?? + ?…+ 
?? ?? ?? ?? 0
?? ?? 
where n ? N 
or  (?? + ?? )
?? = ?
?? =0
?? ? 
?? ?? ?? ?? ?? -?? ?? ?? 
Note : If we put ?? = 1 and ?? = ?? in the above binomial expansion, then 
or  (1+ ?? )
?? = 
?? ?? 0
+ 
?? ?? 1
?? + 
?? ?? 2
?? 2
+ ?+ 
?? ?? ?? ?? ?? + ?+ 
?? ?? ?? ?? ?? 
or  (1+ ?? )
?? = ?
?? =0
?? ? 
?? ?? ?? ?? ?? 
Problem 1 : Expand the following binomials : 
(i)  (?? + v2)
5
 
(ii)  (1-
3?? 2
2
)
4
 
Solution : 
(i)  (?? + v2)
5
= 
5
?? 0
?? 5
+ 
5
?? 1
?? 4
(v2)+ 
5
?? 2
?? 3
(v2)
2
+ 
5
?? 3
?? 2
(v2)
3
+ 
5
?? 4
?? (v2)
4
+
 
5
?? 5
(v2)
5
 
= ?? 5
+ 5v2?? 4
+ 20?? 3
+ 20v2?? 2
+ 20?? + 4v2 
(ii) 
(1 -
3?? 2
2
)
4
 = 
4
?? 0
+ 
4
?? 1
(-
3?? 2
2
) + 
4
?? 2
(-
3?? 2
2
)
2
+ 
4
?? 3
(-
3?? 2
2
)
3
+ 
4
?? 4
(-
3?? 2
2
)
4
 = 1- 6?? 2
27
2
+ ?? 4
-
27
2
?? 6
+
81
16
?? 8
 
Problem 2 : Expand the binomial (
2
?? + ?? )
10
 up to four terms 
Solution :  (
2
?? + ?? )
10
= 
10
?? 0
(
2
?? )
10
+ 
10
?? 1
(
2
?? )
9
?? + 
10
?? 2
(
2
?? )
8
?? 2
+ 
10
?? 3
(
2
?? )
7
?? 3
+ ? 
Observations : 
(i) The number of terms in the binomial expansion (?? + ?? )
?? is ?? + 1. 
(ii) The sum of the indices of ?? and ?? in each term is ?? . 
(iii) The binomial coefficients (  
?? ?? 0
, 
?? ?? 1
……….. 
?? ?? ?? ) of the terms equidistant from the 
beginning and the end are equal, i.e.  
?? ?? 0
= 
?? ?? ?? , 
?? ?? 1
= 
?? ?? ?? -1
 etc. {? 
?? ?? ?? = 
?? ?? ?? -1
} 
(iv) The binomial coefficient can be remembered with the help of the following pascal's 
Triangle (also known as Meru Prastra provided by Pingla) 
Index of the binomial 
The binomial coefficient 
 
Regarding Pascal's Triangle, we note the following : 
(a) Each row of the triangle begins with 1 and ends with 1. 
(b) Any entry in a row is the sum of two entries in the preceding row, one on the 
immediate left and the other on the immediate right. 
Problem 3 : The number of dissimilar terms in the expansion of (1 + ?? 4
- 2?? 2
)
15
 is 
(A) 21 
(B) 31 
(C) 41 
(D) 61 
Solution :  (1 - ?? 2
)
30
 
Therefore number of dissimilar terms = 31 
General term : 
 (?? + ?? )
?? = 
?? ?? 0
?? ?? ?? 0
+ 
?? ?? 1
?? ?? -1
?? 1
+ ?…….+ 
?? ?? ?? ?? ?? -?? ?? ?? + ?…….+ 
?? ?? ?? ?? 0
?? ?? (?? + 1)
?? h
 term is called general term and denoted by ?? ?? +1
.
?? ?? +1
= 
?? ?? ?? ?? ?? -?? ?? ?? 
  
Note : The ?? th 
 term from the end is equal to the (?? - ?? + 2)
th 
 term from the begining, i.e. 
 
?? ?? ?? -?? +1
?? ?? -1
?? ?? -?? +1
 
Problem 4 : Find 
(i) 15
th 
 term of (2?? - 3?? )
20
 
(ii)  4
th 
 term of (
3?? 5
- ?? )
7
 
Page 4


Binomial Theorem 
Binomial expression : 
Any algebraic expression which contains two dissimilar terms is called binomial 
expression. 
For example : ?? + ?? , ?? 2
?? +
1
?? ?? 2
,3- ?? ,v?? 2
+ 1 +
1
(?? 3
+1)
1/3
 etc. 
Terminology used in binomial theorem: 
Factorial notation : n or n ! is pronounced as factorial n and is defined as 
?? ! = {
?? (?? - 1)(?? - 2)…….3 2.1 ;
1 if ?? ? ?? 1 if ?? = 0
 
Note : n! = n.(n- 1)! ; n ? N 
Mathematical meaning of  
?? ?? ?? : The term  
?? ?? ?? denotes number of combinations of ?? 
things choosen from ?? distinct things mathematically,  
?? ?? ?? =
?? !
(?? -?? )! ?? !
,?? ,?? ? ?? ,0 = ?? = ?? 
Note : Other symbols of of  
?? ?? ?? are (
?? ?? ) and ?? (?? ,?? ) . 
Properties related to  
?? ?? ?? : 
(i)  
?? ?? ?? = 
?? ?? ?? -?? 
Note: If  
?? ?? ?? = 
?? ?? ?? ? Either ?? = ?? or ?? + ?? = ?? 
(ii)   
?? ?? ?? + 
?? ?? ?? -1
= 
?? +1
?? ?? 
(iii) 
 
?? ?? ?? 
?? ?? ?? -1
=
?? -?? +1
?? 
(iv)   
?? ?? ?? =
?? ?? 
?? -1
?? ?? -1
=
?? (?? -1)
?? (?? -1)
 
?? -2
?? ?? -2
= =
n(n-1)(n-2)………(n-(r-1))
r(r-1)(r-2)…….2.1
 
(v) If n and r are relatively prime, then  
n
C
r
 is divisible by n. But converse is not 
necessarily true. 
Statement of binomial theorem : 
(?? + ?? )
?? = 
?? ?? 0
?? ?? ?? 0
+ 
?? ?? 1
?? ?? -1
?? 1
+ 
?? ?? 2
?? ?? -2
?? 2
+ ?+ 
?? ?? ?? ?? ?? -?? ?? ?? + ?…+ 
?? ?? ?? ?? 0
?? ?? 
where n ? N 
or  (?? + ?? )
?? = ?
?? =0
?? ? 
?? ?? ?? ?? ?? -?? ?? ?? 
Note : If we put ?? = 1 and ?? = ?? in the above binomial expansion, then 
or  (1+ ?? )
?? = 
?? ?? 0
+ 
?? ?? 1
?? + 
?? ?? 2
?? 2
+ ?+ 
?? ?? ?? ?? ?? + ?+ 
?? ?? ?? ?? ?? 
or  (1+ ?? )
?? = ?
?? =0
?? ? 
?? ?? ?? ?? ?? 
Problem 1 : Expand the following binomials : 
(i)  (?? + v2)
5
 
(ii)  (1-
3?? 2
2
)
4
 
Solution : 
(i)  (?? + v2)
5
= 
5
?? 0
?? 5
+ 
5
?? 1
?? 4
(v2)+ 
5
?? 2
?? 3
(v2)
2
+ 
5
?? 3
?? 2
(v2)
3
+ 
5
?? 4
?? (v2)
4
+
 
5
?? 5
(v2)
5
 
= ?? 5
+ 5v2?? 4
+ 20?? 3
+ 20v2?? 2
+ 20?? + 4v2 
(ii) 
(1 -
3?? 2
2
)
4
 = 
4
?? 0
+ 
4
?? 1
(-
3?? 2
2
) + 
4
?? 2
(-
3?? 2
2
)
2
+ 
4
?? 3
(-
3?? 2
2
)
3
+ 
4
?? 4
(-
3?? 2
2
)
4
 = 1- 6?? 2
27
2
+ ?? 4
-
27
2
?? 6
+
81
16
?? 8
 
Problem 2 : Expand the binomial (
2
?? + ?? )
10
 up to four terms 
Solution :  (
2
?? + ?? )
10
= 
10
?? 0
(
2
?? )
10
+ 
10
?? 1
(
2
?? )
9
?? + 
10
?? 2
(
2
?? )
8
?? 2
+ 
10
?? 3
(
2
?? )
7
?? 3
+ ? 
Observations : 
(i) The number of terms in the binomial expansion (?? + ?? )
?? is ?? + 1. 
(ii) The sum of the indices of ?? and ?? in each term is ?? . 
(iii) The binomial coefficients (  
?? ?? 0
, 
?? ?? 1
……….. 
?? ?? ?? ) of the terms equidistant from the 
beginning and the end are equal, i.e.  
?? ?? 0
= 
?? ?? ?? , 
?? ?? 1
= 
?? ?? ?? -1
 etc. {? 
?? ?? ?? = 
?? ?? ?? -1
} 
(iv) The binomial coefficient can be remembered with the help of the following pascal's 
Triangle (also known as Meru Prastra provided by Pingla) 
Index of the binomial 
The binomial coefficient 
 
Regarding Pascal's Triangle, we note the following : 
(a) Each row of the triangle begins with 1 and ends with 1. 
(b) Any entry in a row is the sum of two entries in the preceding row, one on the 
immediate left and the other on the immediate right. 
Problem 3 : The number of dissimilar terms in the expansion of (1 + ?? 4
- 2?? 2
)
15
 is 
(A) 21 
(B) 31 
(C) 41 
(D) 61 
Solution :  (1 - ?? 2
)
30
 
Therefore number of dissimilar terms = 31 
General term : 
 (?? + ?? )
?? = 
?? ?? 0
?? ?? ?? 0
+ 
?? ?? 1
?? ?? -1
?? 1
+ ?…….+ 
?? ?? ?? ?? ?? -?? ?? ?? + ?…….+ 
?? ?? ?? ?? 0
?? ?? (?? + 1)
?? h
 term is called general term and denoted by ?? ?? +1
.
?? ?? +1
= 
?? ?? ?? ?? ?? -?? ?? ?? 
  
Note : The ?? th 
 term from the end is equal to the (?? - ?? + 2)
th 
 term from the begining, i.e. 
 
?? ?? ?? -?? +1
?? ?? -1
?? ?? -?? +1
 
Problem 4 : Find 
(i) 15
th 
 term of (2?? - 3?? )
20
 
(ii)  4
th 
 term of (
3?? 5
- ?? )
7
 
Solution : 
(i)  ?? 14+1
= 
20
?? 14
(2?? )
6
(-3?? )
14
= 
20
?? 14
2
6
3
14
?? 6
· ?? 14
 
(ii)  ?? 3+1
= 
7
?? 3
(
3?? 5
)
4
(-?? )
3
= 
-7
?? 3
(
3
5
)
4
?? 4
?? 3
 
Problem 5 : Find the number of rational terms in the expansion of (2
1
3
+ 3
1
5)
600
 
Solution : The general term in the expansion of (2
1
3
+ 3
1
5)
600
 is 
T
?? +1
 = 
600
?? ?? (2
1
3
)
600-?? (3
1
5
)
?? = 
600
?? ?? 2
600-?? 3
3
?? 5
 
The above term will be rational if exponent of 3 and 2 are integers 
It means 
600-?? 3
 and 
?? 5
 must be integers. 
The possible set of values of ?? is {0,15,30,45 
Hence, number of rational terms is 41 
Middle term(s) : 
(a) If n is even, there is only one middle term, which is (
n+2
2
)
th 
 term. 
(b) If n is odd, there are two middle terms, which are (
n+1
2
)
th 
 and (
n+1
2
+ 1)
th 
 terms. 
Problem 6 : Find the middle term(s) in the expansion of 
 (i)  (1 + 2?? )
12
 (ii)  (2?? -
?? 2
2
)
11
 
Solution : (i)  (1 + 2?? )
12
 
Here, ?? is even, therefore middle term is (
12+2
2
) th term. 
It means T
7
 is middle term T
7
= 
12
C
6
(2x)
6
 
(ii) 
(2?? -
?? 2
2
)
11
 
Page 5


Binomial Theorem 
Binomial expression : 
Any algebraic expression which contains two dissimilar terms is called binomial 
expression. 
For example : ?? + ?? , ?? 2
?? +
1
?? ?? 2
,3- ?? ,v?? 2
+ 1 +
1
(?? 3
+1)
1/3
 etc. 
Terminology used in binomial theorem: 
Factorial notation : n or n ! is pronounced as factorial n and is defined as 
?? ! = {
?? (?? - 1)(?? - 2)…….3 2.1 ;
1 if ?? ? ?? 1 if ?? = 0
 
Note : n! = n.(n- 1)! ; n ? N 
Mathematical meaning of  
?? ?? ?? : The term  
?? ?? ?? denotes number of combinations of ?? 
things choosen from ?? distinct things mathematically,  
?? ?? ?? =
?? !
(?? -?? )! ?? !
,?? ,?? ? ?? ,0 = ?? = ?? 
Note : Other symbols of of  
?? ?? ?? are (
?? ?? ) and ?? (?? ,?? ) . 
Properties related to  
?? ?? ?? : 
(i)  
?? ?? ?? = 
?? ?? ?? -?? 
Note: If  
?? ?? ?? = 
?? ?? ?? ? Either ?? = ?? or ?? + ?? = ?? 
(ii)   
?? ?? ?? + 
?? ?? ?? -1
= 
?? +1
?? ?? 
(iii) 
 
?? ?? ?? 
?? ?? ?? -1
=
?? -?? +1
?? 
(iv)   
?? ?? ?? =
?? ?? 
?? -1
?? ?? -1
=
?? (?? -1)
?? (?? -1)
 
?? -2
?? ?? -2
= =
n(n-1)(n-2)………(n-(r-1))
r(r-1)(r-2)…….2.1
 
(v) If n and r are relatively prime, then  
n
C
r
 is divisible by n. But converse is not 
necessarily true. 
Statement of binomial theorem : 
(?? + ?? )
?? = 
?? ?? 0
?? ?? ?? 0
+ 
?? ?? 1
?? ?? -1
?? 1
+ 
?? ?? 2
?? ?? -2
?? 2
+ ?+ 
?? ?? ?? ?? ?? -?? ?? ?? + ?…+ 
?? ?? ?? ?? 0
?? ?? 
where n ? N 
or  (?? + ?? )
?? = ?
?? =0
?? ? 
?? ?? ?? ?? ?? -?? ?? ?? 
Note : If we put ?? = 1 and ?? = ?? in the above binomial expansion, then 
or  (1+ ?? )
?? = 
?? ?? 0
+ 
?? ?? 1
?? + 
?? ?? 2
?? 2
+ ?+ 
?? ?? ?? ?? ?? + ?+ 
?? ?? ?? ?? ?? 
or  (1+ ?? )
?? = ?
?? =0
?? ? 
?? ?? ?? ?? ?? 
Problem 1 : Expand the following binomials : 
(i)  (?? + v2)
5
 
(ii)  (1-
3?? 2
2
)
4
 
Solution : 
(i)  (?? + v2)
5
= 
5
?? 0
?? 5
+ 
5
?? 1
?? 4
(v2)+ 
5
?? 2
?? 3
(v2)
2
+ 
5
?? 3
?? 2
(v2)
3
+ 
5
?? 4
?? (v2)
4
+
 
5
?? 5
(v2)
5
 
= ?? 5
+ 5v2?? 4
+ 20?? 3
+ 20v2?? 2
+ 20?? + 4v2 
(ii) 
(1 -
3?? 2
2
)
4
 = 
4
?? 0
+ 
4
?? 1
(-
3?? 2
2
) + 
4
?? 2
(-
3?? 2
2
)
2
+ 
4
?? 3
(-
3?? 2
2
)
3
+ 
4
?? 4
(-
3?? 2
2
)
4
 = 1- 6?? 2
27
2
+ ?? 4
-
27
2
?? 6
+
81
16
?? 8
 
Problem 2 : Expand the binomial (
2
?? + ?? )
10
 up to four terms 
Solution :  (
2
?? + ?? )
10
= 
10
?? 0
(
2
?? )
10
+ 
10
?? 1
(
2
?? )
9
?? + 
10
?? 2
(
2
?? )
8
?? 2
+ 
10
?? 3
(
2
?? )
7
?? 3
+ ? 
Observations : 
(i) The number of terms in the binomial expansion (?? + ?? )
?? is ?? + 1. 
(ii) The sum of the indices of ?? and ?? in each term is ?? . 
(iii) The binomial coefficients (  
?? ?? 0
, 
?? ?? 1
……….. 
?? ?? ?? ) of the terms equidistant from the 
beginning and the end are equal, i.e.  
?? ?? 0
= 
?? ?? ?? , 
?? ?? 1
= 
?? ?? ?? -1
 etc. {? 
?? ?? ?? = 
?? ?? ?? -1
} 
(iv) The binomial coefficient can be remembered with the help of the following pascal's 
Triangle (also known as Meru Prastra provided by Pingla) 
Index of the binomial 
The binomial coefficient 
 
Regarding Pascal's Triangle, we note the following : 
(a) Each row of the triangle begins with 1 and ends with 1. 
(b) Any entry in a row is the sum of two entries in the preceding row, one on the 
immediate left and the other on the immediate right. 
Problem 3 : The number of dissimilar terms in the expansion of (1 + ?? 4
- 2?? 2
)
15
 is 
(A) 21 
(B) 31 
(C) 41 
(D) 61 
Solution :  (1 - ?? 2
)
30
 
Therefore number of dissimilar terms = 31 
General term : 
 (?? + ?? )
?? = 
?? ?? 0
?? ?? ?? 0
+ 
?? ?? 1
?? ?? -1
?? 1
+ ?…….+ 
?? ?? ?? ?? ?? -?? ?? ?? + ?…….+ 
?? ?? ?? ?? 0
?? ?? (?? + 1)
?? h
 term is called general term and denoted by ?? ?? +1
.
?? ?? +1
= 
?? ?? ?? ?? ?? -?? ?? ?? 
  
Note : The ?? th 
 term from the end is equal to the (?? - ?? + 2)
th 
 term from the begining, i.e. 
 
?? ?? ?? -?? +1
?? ?? -1
?? ?? -?? +1
 
Problem 4 : Find 
(i) 15
th 
 term of (2?? - 3?? )
20
 
(ii)  4
th 
 term of (
3?? 5
- ?? )
7
 
Solution : 
(i)  ?? 14+1
= 
20
?? 14
(2?? )
6
(-3?? )
14
= 
20
?? 14
2
6
3
14
?? 6
· ?? 14
 
(ii)  ?? 3+1
= 
7
?? 3
(
3?? 5
)
4
(-?? )
3
= 
-7
?? 3
(
3
5
)
4
?? 4
?? 3
 
Problem 5 : Find the number of rational terms in the expansion of (2
1
3
+ 3
1
5)
600
 
Solution : The general term in the expansion of (2
1
3
+ 3
1
5)
600
 is 
T
?? +1
 = 
600
?? ?? (2
1
3
)
600-?? (3
1
5
)
?? = 
600
?? ?? 2
600-?? 3
3
?? 5
 
The above term will be rational if exponent of 3 and 2 are integers 
It means 
600-?? 3
 and 
?? 5
 must be integers. 
The possible set of values of ?? is {0,15,30,45 
Hence, number of rational terms is 41 
Middle term(s) : 
(a) If n is even, there is only one middle term, which is (
n+2
2
)
th 
 term. 
(b) If n is odd, there are two middle terms, which are (
n+1
2
)
th 
 and (
n+1
2
+ 1)
th 
 terms. 
Problem 6 : Find the middle term(s) in the expansion of 
 (i)  (1 + 2?? )
12
 (ii)  (2?? -
?? 2
2
)
11
 
Solution : (i)  (1 + 2?? )
12
 
Here, ?? is even, therefore middle term is (
12+2
2
) th term. 
It means T
7
 is middle term T
7
= 
12
C
6
(2x)
6
 
(ii) 
(2?? -
?? 2
2
)
11
 
Here, n is odd therefore, middle terms are (
11+1
2
) th &(
11+1
2
+ 1) th. 
It means T
6
& T
7
 are middle terms 
?? 6
= 
11
?? 5
(2?? )
6
(-
?? 2
2
)
5
= -2 
11
?? 5
?? 16
? ?? 7
= 
11
?? 6
(2?? )
5
(-
?? 2
2
)
6
=
 
11
?? 6
2
?? 17
 
Problem 7 : Find term which is independent of ?? in (?? 2
-
1
?? 6
)
16
 
Solution :  ?? ?? +1
= 
16
?? ?? (?? 2
)
16-?? (-
1
?? 6
)
?? 
For term to be independent of ?? , exponent of ?? should be 0 
32- 2?? = 6?? ? ?? = 4 ? ?? 5
 is independent of ?? . 
Numerically greatest term in the 
expansion of (?? + b)
?? ,?? ? ?? 
Binomial expansion of (?? + ?? )
?? is as follows : - 
(?? + ?? )
?? = 
?? ?? 0
?? ?? ?? 0
+ 
?? ?? 1
?? ?? -1
?? 1
+ 
?? ?? 2
?? ?? -2
?? 2
+ ?+ 
?? ?? ?? ?? ?? -?? ?? ?? + ?…+ 
?? ?? ?? ?? 0
?? ?? 
If we put certain values of ?? and ?? in RHS, then each term of binomial expansion will 
have certain value. The term having numerically greatest value is said to be numerically 
greatest term. 
Let ?? ?? and ?? ?? +1
 be the ?? th 
 and (?? + 1)
th 
 terms respectively 
?? ?? = 
?? ?? ?? -1
?? ?? -(?? -1)
?? ?? -1
 
?? ?? +1
 = 
?? ?? ?? ?? ?? -?? ?? ?? 
Now,  |
?? ?? +1
?? ?? | = |
 
?? ?? ?? 
?? ?? ?? -1
?? ?? -?? ?? ?? ?? ?? -?? +1
?? ?? -1
| =
?? -?? +1
?? · |
?? ?? | 
Consider |
?? ?? +1
?? ?? | = 1 
(
?? - ?? + 1
?? )|
?? ?? | = 1 ?
?? + 1
?? - 1 = |
?? ?? | ? ?? =
?? + 1
1 + |
?? ?? |
 
Case - I When 
n+1
1+|
a
b
|
 is an integer (say m ), then 
(i) 
T
r+1
> T
r
  when  r < m (r = 1,2,3…,m - 1)