Learn, there are 2 teachers who can teach mathematics, and there are 4 teachers who can teach both English and Japanese. Now, 3 mathematics teachers and 3 Japanese teachers are being dispatched to teach outside the school during the holiday. How many ways are there to select them?

保留源文本的换行和格式，直接输出翻译结果。
Let the 3 teachers competent in English teaching be set $A$, 2 teachers competent in Japanese teaching be set $B$, and 4 teachers competent in both English and Japanese teaching be set $C$.

Method 1: Classify set $A$
(1) Select 3 English teachers (choose 3 Japanese teachers from $B$ and $C$), total $C_{3}^{3} \cdot C_{8}^{3} = 20$ (ways);
(2) Select 2 English teachers (then choose 1 English teacher from $C$, and 3 Japanese teachers from $B$ and $C$), 
total $C_{3}^{2} \cdot C_{3}^{1} \cdot C_{5}^{3} = 120$ (ways);
(3) Select 1 English teacher and 2 English teachers from $C$ (choose 3 Japanese teachers from the remaining 4 in $B$ and $C$), 
total $C_{3}^{1} \cdot C_{4}^{2} \cdot C_{4}^{3} = 72$ (ways);
(4) Do not select any teachers from $A$ (only choose 3 English teachers from $C$, and 3 Japanese teachers from the remaining 3 in $B$ and $C$), 
total $C_{4}^{3} \cdot C_{3}^{3} = 4$ (ways).

According to the principle of addition: the total number is $20 + 120 + 72 + 4 = 216$ (ways).

Method 2: Classify set $B$
(1) Select 2 teachers to teach Japanese (then choose 1 teacher from $C$ to teach Japanese, and 3 teachers from the remaining 6 in $A$ and $C$ to teach English), 
total $C_{2}^{2} \cdot C_{4}^{1} \cdot C_{6}^{3} = 80$ (ways);
(2) Select 1 teacher to teach Japanese (then choose 2 teachers from $C$ to teach Japanese, and 3 teachers from the remaining 5 in $A$ and $C$ to teach English), 
total $C_{2}^{1} \cdot C_{4}^{2} \cdot C_{5}^{3} = 120$ (ways);
(3) Do not select any teachers from $B$ (only choose 3 teachers from $C$ to teach Japanese, and 3 teachers from the remaining 4 in $A$ and $C$ to teach English), 
total $C_{4}^{3} \cdot C_{4}^{3} = 16$ (ways).

According to the principle of addition: the total number is $80 + 120 + 16 = 216$ (ways).

This problem can also be solved by classifying set $C$.
In permutation and combination problems, errors such as missing solutions or overlaps often occur, or the conditions may seem messy and difficult to handle. One reason for this is the inability to correctly classify the objects involved in the problem.
(Author's unit: Li Jiashen High School, Baodi County)