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National and Regional Contests
Moldova Contests
Chisinau City MO
1977 Chisinau City MO
140
140
Part of
1977 Chisinau City MO
Problems
(1)
Chisinau MO p140 1977 IX sum nC_{n}^{n} =n x 2 ^{n-1}
Source:
3/17/2021
Prove the identities:
C
n
1
+
2
C
n
2
+
3
C
n
3
+
.
.
.
+
n
C
n
n
=
n
⋅
2
n
−
1
C_{n}^{1}+2C_{n}^{2}+3C_{n}^{3}+...+nC_{n}^{n}=n\cdot 2 ^{n-1}
C
n
1
+
2
C
n
2
+
3
C
n
3
+
...
+
n
C
n
n
=
n
⋅
2
n
−
1
C
n
1
−
2
C
n
2
+
3
C
n
3
+
.
.
.
−
(
−
1
)
n
−
1
n
C
n
n
=
0
C_{n}^{1}-2C_{n}^{2}+3C_{n}^{3}+...-(-1)^{n-1}nC_{n}^{n}=0
C
n
1
−
2
C
n
2
+
3
C
n
3
+
...
−
(
−
1
)
n
−
1
n
C
n
n
=
0
combinatorics
Combinations