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Byju's Answer
Standard XIII
Mathematics
Sum of Coefficients of All Terms
If 1+xn=C0+C1...
Question
If
(
1
+
x
)
n
=
C
0
+
C
1
x
+
…
.
.
+
C
n
x
n
, then the value of
∑
∑
0
≤
r
<
s
≤
n
C
r
C
s
is equal to
A
1
2
[
2
2
n
−
2
n
C
n
]
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B
1
4
[
2
2
n
−
2
n
C
n
]
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C
1
2
[
2
2
n
+
2
n
C
n
]
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D
1
2
[
2
n
−
2
n
C
n
]
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Solution
The correct option is
A
1
2
[
2
2
n
−
2
n
C
n
]
We have,
n
∑
r
=
0
n
∑
s
=
0
C
r
C
s
=
(
n
∑
r
=
0
C
2
r
)
+
2
∑
∑
0
≤
r
<
s
≤
n
C
r
C
s
⇒
2
2
n
=
2
n
C
n
+
2
∑
∑
0
≤
r
<
s
≤
n
C
r
C
s
⇒
∑
∑
0
≤
r
<
s
≤
n
C
r
C
s
=
1
2
[
2
2
n
−
2
n
C
n
]
Suggest Corrections
0
Similar questions
Q.
If
(
1
+
x
)
n
=
C
0
+
C
1
x
+
.
.
.
+
C
n
x
n
, then the value of
∑
0
≤
r
<
∑
s
≤
n
(
C
r
+
C
s
)
is equal to
Q.
If
c
0
,
c
1
,
c
2
,
.
.
.
.
.
.
.
c
n
denote the coefficients in the expansion of
(
1
+
x
)
n
, prove that
c
0
c
r
+
c
1
c
r
+
1
+
c
2
c
r
+
2
+
.
.
.
.
+
c
n
−
r
c
n
=
|
2
n
–
–
–
|
n
−
r
–
––––
–
|
n
+
r
–
––––
–
.
Q.
If
(
1
+
x
)
n
=
n
∑
r
=
0
C
r
x
r
, the value of
C
0
+
(
C
0
+
C
1
)
+
(
C
0
+
C
1
+
C
2
)
+
⋯
(
C
0
+
C
1
+
C
2
+
⋯
+
C
n
−
1
)
equals
Q.
If
(
1
+
x
)
n
=
n
∑
r
=
0
n
C
r
then solve
(
C
0
+
C
1
C
0
)
(
C
1
+
C
2
C
1
)
(
C
2
+
C
3
C
2
)
.
.
.
.
.
.
.
.
.
(
C
n
−
1
+
C
n
C
n
−
1
)
Q.
If
(
1
+
x
)
n
=
C
0
+
C
1
x
+
C
2
x
2
+
.
.
.
.
.
.
.
.
.
.
.
C
n
x
n
then
C
0
.
C
r
+
C
1
.
C
r
+
C
2
.
C
r
+
2
+
.
.
.
.
.
.
.
+
C
n
−
r
.
C
n
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