Solution 1: Number the 8 targets in the order they are shot.
First, consider the leftmost column. The number of ways to shoot down these 3 targets is $C_{8}^{3}$;
Next, consider the middle column. The number of ways to shoot down these 2 targets is $C_{5}^{2}$;
Solution 2: Number the 8 targets in the order they are shot, and divide the corresponding numbers into 3 groups, each with 3, 2, and 3 numbers, respectively.

For each group of numbers, we take the smallest number and shoot the lowest clay ball in the corresponding column first according to the number; we take the largest number and shoot the highest clay ball in the corresponding column last according to the number.

Thus, this can be seen as a permutation problem with a finite number of repeated elements, so the number of permutations is $\overline{3!} \frac{8!}{2!} 3!=560$ ways. Once the order of shooting the targets in the first two columns is determined, the order of shooting the targets in the rightmost column is uniquely determined. According to the principle of step-by-step counting, the number of different orders to shoot down these 8 targets is $C_{8}^{3} \times C_{5}^{2}=560$ ways.
Example 2 In a shooting competition, there are 8 clay targets hanging as shown in Figure 18-1. A marksman shoots the targets according to the following rules:
(1) First, choose a column from which a target will be shot down;
(2) Then, in the selected column, shoot down the lowest remaining target.
How many different sequences are there to shoot down these 8 targets?