Example 12 A deck of three-color cards, with a total of 32 cards, where each of the red, yellow, and blue cards are 10 each, numbered $1, 2, \cdots, 10$; there are also a big and a small joker, each numbered 0. From this deck, several cards are drawn, and then the score is calculated according to the following rules: each card numbered $k$ is counted as $2^{k}$ points. If the sum of their scores is 2004, these cards are called a "good" card group. Try to find the number of "good" card groups.
(2004 National Girls' Olympiad)
For $n \in\{1,2, \cdots, 2004\}$, let $a_{n}$ denote the number of card sets with a total score of $n$. Then $a_{n}$ is equal to the coefficient of $x^{n}$ in the expansion of the function $f(x)=\left(1+x^{2}\right)^{2}\left(1+x^{2^{1}}\right)^{3} \cdots\left(1-x^{2^{10}}\right)^{3}$ (with the convention $|x|<1$).
Since $f(x)=\frac{1}{1+x}\left[\left(1+x^{2^{0}}\right)\left(1+x^{2^{1}}\right) \cdots\left(1+x^{2^{10}}\right]^{3}\right.$
$$
=\frac{1}{(1+x)(1-x)^{3}}\left(1-x^{2^{11}}\right)^{3}=\frac{1}{\left(1-x^{2}\right)(1-x)^{2}}\left(1-x^{2^{11}}\right)^{3} \text {, }
$$

and $n \leqslant 2004<2^{11}$, so $a_{n}$ is equal to the coefficient of $x^{n}$ in the expansion of $\frac{1}{\left(1-x^{2}\right)(1-x)^{2}}$. Since
$$
\begin{array}{l}
\frac{1}{\left(1-x^{2}\right)(1-x)^{2}}=\frac{1}{1-x^{2}} \cdot \frac{1}{(1-x)^{2}} \\
=\left(1+x^{2}+x^{4}+\cdots+x^{2 k}+\cdots\right)\left[1+2 x+3 x^{2}+\cdots+(2 k+1) x^{2 k}+\right. \\
\text { ...]. } \\
\end{array}
$$

Thus, the coefficient of $x^{2 k}$ is $a_{2 k}=1+3+5+\cdots+(2 k+1)=(k+1)^{2}, k=1,2, \cdots$.
Therefore, the number of "good" card sets is $a_{2004}=1003^{2}=1006009$.