【Example 9】There are 10 boys and 10 girls lining up in two rows to dance together, (1) if boys and girls stand in separate rows; (2) if they stand in two rows without considering gender, with 10 people in each row; (3) if boys and girls stand in separate rows, not considering the order within each row, only considering which two people are paired. How many different situations are there in each case?
(1) There are $A_{10}^{10}$ ways to arrange the boys and $A_{10}^{10}$ ways to arrange the girls. Considering the relative positions between the two lines, we know there are
$$
A_{10}^{10} \cdot A_{10}^{10} \cdot A_{2}^{2}=(10!)^{2} \cdot 2!=2 \cdot(10!)^{2}
$$

different cases.
(2) It is only necessary to place 20 people in 20 different positions, so there are $A_{20}^{20}=20$! different cases.
(3) Consider the pairing of boys and girls one by one: there are 10 possible partners for boy A; 9 possible partners for boy B; ...; and 1 possible partner for boy J. Additionally, there is the issue of the positions between the two lines, so there are $2 \cdot 10$! different cases.