OliForum Matematico

for n=3, we have 7 teams: 1,2,3,12,13,23,123
[...]
second posibility: 1 , 2 , 3 , 1 2, 1 3, 2 3,1 2 3
[...]we have 24 posibilities
We do not. Be careful: second possibility above does not verify the hypothesis. Take A = \{ 1, 2 \} and B = \{3\} .

Then the manager of A \cup B is 2 , that is ...

we have $ 2^n-1 $ teams.
so in each we must choose a manager.
for n=3, we have 7 teams: 1,2,3,12,13,23,123
fisrst posibility: 1,2,3,12,13,23,123
second posibility: 1,2,3,12,13,23,123
third posibility: 1,2,3,12,13,23,123
go on, we have 24 posibilities

I think n! is bad answer.
if n=3, we have 24 posibilities.