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Moldova Team Selection Test
2012 Moldova Team Selection Test
12
Identity
Identity
Source:
January 7, 2016
combinatorics
Problem Statement
Let
k
∈
N
k \in \mathbb{N}
k
∈
N
. Prove that
(
k
0
)
⋅
(
x
+
k
)
k
−
(
k
1
)
⋅
(
x
+
k
−
1
)
k
+
.
.
.
+
(
−
1
)
k
⋅
(
k
k
)
⋅
x
k
=
k
!
,
∀
k
∈
R
\binom{k}{0} \cdot (x+k)^k - \binom{k}{1} \cdot (x+k-1)^k+...+(-1)^k \cdot \binom{k}{k} \cdot x^k=k! ,\forall k \in \mathbb{R}
(
0
k
)
⋅
(
x
+
k
)
k
−
(
1
k
)
⋅
(
x
+
k
−
1
)
k
+
...
+
(
−
1
)
k
⋅
(
k
k
)
⋅
x
k
=
k
!
,
∀
k
∈
R
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