Example 5 A deck of tri-color cards consists of 32 cards, with 10 cards of each color: red, yellow, and blue, numbered $1, 2, 3, \cdots, 10$; there are also a big joker and a small joker, each numbered 0. From this deck, several cards are drawn, and their scores are calculated according to the following rule: each card with a number $k$ is worth $2^{k}$ points. If the sum of their scores equals 2004, these cards are called a "good" card group. Find the number of "good" card groups.
(2004 3rd China Girls Mathematical Olympiad)
Let $a_{n}$ denote the number of card groups with a sum of scores equal to $n$. Thus, the generating function for $a_{n}$ is
$$
\begin{aligned}
f(x) & =\sum_{n=0}^{\infty} a_{n} x^{n}=\left(1+x^{2^{0}}\right)^{2}\left(1+x^{2^{1}}\right)^{3}\left(1+x^{2^{2}}\right)^{3} \cdots\left(1+x^{2^{10}}\right)^{3} \\
& =\frac{1}{1+x}\left\{\left(1+x^{2^{0}}\right)\left(1+x^{2^{1}}\right)\left(1+x^{2^{2}}\right) \cdots\left(1+x^{2^{10}}\right)\right\}^{3} \\
& =\frac{1}{(1+x)(1-x)^{3}} \cdot\left(1-x^{2^{10}}\right)^{3} .
\end{aligned}
$$

Since $2004<2^{11}$, $a_{2004}$ is equal to the coefficient of $x^{2004}$ in the expansion of $\frac{1}{(1+x)(1-x)^{3}}$. And
$$
\frac{1}{(1+x)(1-x)^{3}}=\frac{1}{\left(1-x^{2}\right)(1-x)^{2}}=\sum_{i=0}^{\infty} x^{2 i} \sum_{j=0}^{\infty} C_{j+1}^{1} x^{j},
$$

Thus, the coefficient of $x^{2004}$ is
$$
\begin{aligned}
a_{2004} & =C_{2005}^{1}+C_{2003}^{1}+\cdots+C_{3}^{1}+C_{1}^{1} \\
& =2005+2003+\cdots+3+1=1003^{2}=1006009 .
\end{aligned}
$$

Therefore, the number of "good" card groups is 1006009.