Example 2: Now we have 1 one-jiao note, 1 wu-jiao note, 1 five-jiao note, 5 one-yuan notes, and 2 five-yuan notes. How many different amounts of money (excluding the case of not paying) can be paid?
Solution: Using 1 jiao coins can form 0 jiao, 1 jiao, 2 different amounts;
Using 2 jiao coins can form 0 jiao, 2 jiao, 2 different amounts;
Using 5 jiao coins can form 0 jiao, 5 jiao, 2 different amounts;
Using 1 yuan coins can form 0 yuan, 1 yuan, 2 yuan, 3 yuan, 4 yuan, 5 yuan, a total of 6 different amounts;
Using 5 yuan can form 0 yuan, 5 yuan, 10 yuan, a total of 3 different amounts.
Notice that the payment amounts of 5, 5.1, 5.2, 5.3, 5.5, 5.6, 5.7, 5.8 yuan, 10, 10.1, 10.2, 10.3, 10.5, 10.6, 10.7, 10.8 yuan each have 2 methods, so the number of different payment methods is
$$
2 \times 2 \times 2 \times 6 \times 3 - 16 - 1 = 127 \text{ methods. }
$$