2 Let $A_{i}$ be a subset of $M=\{1,2, \cdots, 10\}$, and $\left|A_{i}\right|=5(i=1,2, \cdots, k)$, $\left|A_{i} \cap A_{j}\right| \leqslant 2(1 \leqslant i<j \leqslant k)$. Find the maximum value of $k$. (1994 China National Training Team
2. Since $\sum_{i=1}^{10} m_{i}=\sum_{i=1}^{k} 5=5 k$, by Cauchy's inequality, we have $2 \sum_{1 \leqslant i<j \leqslant k} \mid A_{i} \cap A_{j} \mid=2 \sum_{i=1}^{10} \mathrm{C}_{m_{i}}^{2}=\sum_{i=1}^{10} m_{i}^{2}-\sum_{i=1}^{10} m_{i} \geqslant \frac{\left(\sum_{i=1}^{10} m_{i}\right)^{2}}{\sum_{i=1}^{10} 1^{2}}-\sum_{i=1}^{10} m_{i}=\frac{25 k^{2}}{10}-5 k$. Also, by the condition, $\sum_{1 \leqslant i \leqslant k}\left|A_{i} \cap A_{j}\right| \leqslant \sum_{1 \leqslant i<j \leqslant k} 2=2 C_{k}^{2}=k^{2}-k$. Combining the two inequalities, we get $k \leqslant 6$. When $k=6$, the 6 sets: $\{1,2,3,4,5\}, \{3,5,7,8,9\}, \{1,2,6,7,8\}, \{1,3,6,9,10\}, \{2,4,7,9,10\}, \{4,5,6,8,10\}$, satisfy the conditions, hence the maximum value of $k$ is 6.