Binomial Theorem Class 11 Notes Maths Chapter 7

 Page 1


 
KEY POINTS 
? If n is a natural number and a, b are any numbers. 
1 2 2
0 1 2
0
( ) ............
,
n n n n n n n n n
n
n
n n r r
r
r
a b c a c a b c a b c b
c a b n N
? ?
?
?
? ? ? ? ? ?
? ?
?
? Tr + 1 = General term 
0
n n r r
r
c a b r n
?
? ? ? 
? Total number of terms in (a + b)
n
 is (n + 1). 
? If n is even, then in the expansion of ( )
n
a b ? , middle term is 
1
2
th
n ? ?
?
? ?
? ?
term i,e 
2
2
th
n ? ? ?
? ?
? ?
terms. 
? If n is odd, then in the expansion of (a + b)
n
, middle terms are 
1
2
th
n ? ? ?
? ?
? ?
 and 
3
2
th
n ? ? ?
? ?
? ?
? In (a + b)
n
, r
th
 term from the end is same as ( n – r + 2)
th
 term 
from the beginning. 
Page 2


 
KEY POINTS 
? If n is a natural number and a, b are any numbers. 
1 2 2
0 1 2
0
( ) ............
,
n n n n n n n n n
n
n
n n r r
r
r
a b c a c a b c a b c b
c a b n N
? ?
?
?
? ? ? ? ? ?
? ?
?
? Tr + 1 = General term 
0
n n r r
r
c a b r n
?
? ? ? 
? Total number of terms in (a + b)
n
 is (n + 1). 
? If n is even, then in the expansion of ( )
n
a b ? , middle term is 
1
2
th
n ? ?
?
? ?
? ?
term i,e 
2
2
th
n ? ? ?
? ?
? ?
terms. 
? If n is odd, then in the expansion of (a + b)
n
, middle terms are 
1
2
th
n ? ? ?
? ?
? ?
 and 
3
2
th
n ? ? ?
? ?
? ?
? In (a + b)
n
, r
th
 term from the end is same as ( n – r + 2)
th
 term 
from the beginning. 
 
? r
th
 term from the end in (b + a)
n
= r
th
 term from the beginning in (a + b)
n
 
? In (1 + x)
n
, coefficient of x
r
 is 
n
Cr 
? Some particular cases: 
? ?
0
1
n
n
n r
r
r
x c x
?
? ?
?
? ?
0
1 ( 1 )
n
n
r n r
r
r
x c x
?
? ? ?
?
? Some properties of Binomial coefficients: 
n
C0 + 
n
C1 + 
n
C2 + ... + 
n
Cn = 2
n
n
C0 – 
n
C1 + 
n
C2 – 
n
C3 + ... + (–1)
nn
Cn = 0 
n
C0 + 
n
C2 + 
n
C4 + ... = 
n
C1 + 
n
C3 + n
C
5 + ... = 2
n–1