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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_1 x + C_2 x^2 + \dots + C_n x^n
(
1
+
x
)
n
=
C
0
+
C
1
x
+
C
2
x
2
+
⋯
+
C
n
x
n
, prove that
C
0
2
+
C
1
2
2
+
C
2
2
3
+
⋯
+
C
n
2
n
+
1
=
(
2
n
+
1
)
!
(
n
+
1
)
!
2
C_0^2 + \frac{C_1^2}{2} + \frac{C_2^2}{3} + \dots + \frac{C_n^2}{n+1} = \frac{(2n+1)!}{(n+1)!^2}
C
0
2
+
2
C
1
2
+
3
C
2
2
+
⋯
+
n
+
1
C
n
2
=
(
n
+
1
)
!
2
(
2
n
+
1
)!
.
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