The 25 member states of the European Union set up a committee with the following rules: (1) the committee should meet daily; (2) at each meeting, at least one member state should be represented; (3) at any two different meetings, a different set of member states should be represented; and (4) at the $n$ 'th meeting, for every $k<n$, the set of states represented should include at least one state that was represented at the k'th meeting. For how many days can the committee have its meetings?

Answer: At most $2^{24}=16777216$ days.
If one member is always represented, rules 2 and 4 will be fulfilled. There are $2^{24}$ different subsets of the remaining 24 members, so there can be at least $2^{24}$ meetings. Rule 3 forbids complementary sets at two different meetings, so the maximal number of meetings cannot exceed $\frac{1}{2} \cdot 2^{25}=2^{24}$. So the maximal number of meetings for the committee is exactly $2^{24}=16777216$.