3. A school has 1200 students, each of whom is a member of exactly $k$ clubs. It is known that for any 23 students, they are all members of some common club, but no club has all 1200 students as members. Find the minimum possible value of $k$.
3. $k=23$.

First, prove: $k \leqslant 23$.
Let $S_{1}, S_{2}, \cdots, S_{1200}$ represent all the students, and $C_{1}$, $C_{2}, \cdots, C_{24}$ represent different clubs.

Assume student $S_{i}(1 \leqslant i \leqslant 24)$ participates in clubs $C_{1}, \cdots, C_{i-1}$, $C_{i+1}, \cdots, C_{24}$. The remaining students all participate in the same club as student $S_{1}$. In this case, every 23 students participate in some common club, and there is no club that all 1200 students participate in.
Next, prove: $k \geqslant 23$.
Assume student $T$ participates in clubs $M_{1}, M_{2}, \cdots, M_{k}$.
Since all 1200 students do not participate in a common club, for each $1 \leqslant i \leqslant k$, there exists a student $T_{i}$ who does not participate in club $M_{i}$.

Thus, students $T, T_{1}, \cdots, T_{k}$ do not participate in a common club. Therefore, $k \geqslant 23$.
In conclusion, $k=23$.