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Italy Contests
Italy TST
1994 Italy TST
4
\sum_{(E_1,...,E_k) \in E_k }|E_1 \cap ... \cap E_k|
\sum_{(E_1,...,E_k) \in E_k }|E_1 \cap ... \cap E_k|
Source: Italy TST 1994 p4
February 17, 2020
Subsets
algebra
Sets
Problem Statement
Let
X
X
X
be a set of
n
n
n
elements and
k
k
k
be a positive integer. Consider the family
S
k
S_k
S
k
of all
k
k
k
-tuples
(
E
1
,
.
.
.
,
E
k
)
(E_1,...,E_k)
(
E
1
,
...
,
E
k
)
with
E
i
⊆
X
E_i \subseteq X
E
i
⊆
X
for each
i
i
i
. Evaluate the sums
∑
(
E
1
,
.
.
.
,
E
k
)
∈
S
k
∣
E
1
∩
.
.
.
∩
E
k
∣
\sum_{(E_1,...,E_k) \in S_k }|E_1 \cap ... \cap E_k|
∑
(
E
1
,
...
,
E
k
)
∈
S
k
∣
E
1
∩
...
∩
E
k
∣
and
∑
(
E
1
,
.
.
.
,
E
k
)
∈
S
k
∣
E
1
∪
.
.
.
∪
E
k
∣
\sum_{(E_1,...,E_k) \in S_k }|E_1 \cup ... \cup E_k|
∑
(
E
1
,
...
,
E
k
)
∈
S
k
∣
E
1
∪
...
∪
E
k
∣
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