Important Formulas: Binomial Theorem & its Simple Applications

# Important Binomial Theorem & its Simple Applications Formulas for JEE and NEET

``` Page 1

Page # 26
BINOMIAL THEOREM
1. Statement of Binomial theorem :  If a, b ? R and n ? N, then
(a + b)
n
=
n
C
0
a
n
b
0
+
n
C
1
a
n–1
b
1
+
n
C
2
a
n–2
b
2
+...+
n
C
r
a
n–r
b
r
+...+
n
C
n
a
0
b
n
= ?
?
?
n
0 r
r r n
r
n
b a C
2. Properties of Binomial Theorem :
(i) General term :  T
r+1
=
n
C
r
a
n–r
b
r
(ii) Middle term (s) :
(a) If n is even, there is only one middle term,
which is
?
?
?
?
?
? ?
2
2 n
th term.
(b) If n is odd, there are two middle terms,
which are
?
?
?
?
?
? ?
2
1 n
th and
?
?
?
?
?
?
?
?
1
2
1 n
th terms.
Page 2

Page # 26
BINOMIAL THEOREM
1. Statement of Binomial theorem :  If a, b ? R and n ? N, then
(a + b)
n
=
n
C
0
a
n
b
0
+
n
C
1
a
n–1
b
1
+
n
C
2
a
n–2
b
2
+...+
n
C
r
a
n–r
b
r
+...+
n
C
n
a
0
b
n
= ?
?
?
n
0 r
r r n
r
n
b a C
2. Properties of Binomial Theorem :
(i) General term :  T
r+1
=
n
C
r
a
n–r
b
r
(ii) Middle term (s) :
(a) If n is even, there is only one middle term,
which is
?
?
?
?
?
? ?
2
2 n
th term.
(b) If n is odd, there are two middle terms,
which are
?
?
?
?
?
? ?
2
1 n
th and
?
?
?
?
?
?
?
?
1
2
1 n
th terms.
Page # 27
3. Multinomial Theorem :
(x
1
+ x
2
+ x
3
+ ........... x
k
)
n
=
?
???? n r ... r r
k 2 1
k 2 1
! r !... r ! r
! n

k 2 1
r
k
r
2
r
1
x ... x . x
Here total number of terms in the expansion =
n+k–1
C
k–1
4. Application of Binomial Theorem :
If
n
) B A ( ? = ? + f where ? and  n are positive integers, n being odd and
0 < f < 1  then ( ? + f) f = k
n
where A – B
2
= k > 0 and A – B < 1.
If n is an even integer, then ( ? + f) (1 – f) = k
n
5. Properties of Binomial Coefficients :
(i)
n
C
0
+
n
C
1
+
n
C
2
+ ........+
n
C
n
= 2
n
(ii)
n
C
0
–
n
C
1
+
n
C
2
–
n
C
3
+ ............. + (–1)
n

n
C
n
= 0
(iii)
n
C
0
+
n
C
2
+
n
C
4
+ .... =
n
C
1
+
n
C
3
+
n
C
5
+ .... = 2
n–1
(iv)
n
C
r
+
n
C
r–1
=
n+1
C
r
(v)
1 r
n
r
n
C
C
?
=
r
1 r n ? ?
6. Binomial Theorem For Negative Integer Or Fractional Indices
(1 + x)
n
= 1 + nx +
! 2
) 1 n ( n ?
x
2
+
! 3
) 2 n )( 1 n ( n ? ?
x
3
+ .... +
! r
) 1 r n ).......( 2 n )( 1 n ( n ? ? ? ?
x
r
+ ....,| x | < 1.
T
r+1
=
! r
) 1 r n ( )......... 2 n )( 1 n ( n ? ? ? ?
x
r
```

## Mathematics (Maths) for JEE Main & Advanced

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## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

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