Example 4: Given 1 one-yuan note, 1 two-yuan note, 1 five-yuan note, 4 ten-yuan notes, and 2 fifty-yuan notes, how many different amounts of money can be paid using these notes?
(1986 Shanghai Competition Question)
This is a problem of finite repeated combinations. According to the problem, the generating function is $f(x)=(1+x)(1+x^2)(1+x^5)(1+x^{10}+x^{20}+x^{30}+x^{40})(1+x^{50}+x^{100})$. Expanding this, we find that the highest power of $x$ is 148, which indicates that any amount exceeding 148 jiao cannot be paid. However, in reality, amounts of 4 jiao, 9 jiao, 14 jiao, 19 jiao, ..., 144 jiao, totaling 29 different amounts, also cannot be paid. Therefore, the total number of different amounts that can be paid is 119.