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Mathematics
Mathematics
Problem
If
(
1
+
x
)
n
=
C
0
+
C
1
x
+
C
2
x
2
+
⋯
+
C
n
x
n
(1+x)^n = C_0 + C_1x + C_2x^2 + \dots + C_n x^n
(
1
+
x
)
n
=
C
0
+
C
1
x
+
C
2
x
2
+
⋯
+
C
n
x
n
, prove that
(
1
⋅
2
)
C
2
+
(
2
⋅
3
)
C
3
+
⋯
+
(
n
−
1
)
⋅
n
⋅
C
n
=
n
(
n
−
1
)
2
n
−
2
(1 \cdot 2)C_2 + (2 \cdot 3)C_3 + \dots + (n-1) \cdot n \cdot C_n = n(n-1)2^{n-2}
(
1
⋅
2
)
C
2
+
(
2
⋅
3
)
C
3
+
⋯
+
(
n
−
1
)
⋅
n
⋅
C
n
=
n
(
n
−
1
)
2
n
−
2
.
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