Example 12 A deck of tri-color cards consists of 32 cards in total, with 10 cards of each color: red, yellow, and blue, numbered $1,2, \cdots, 10$; there are also a big joker and a small joker, each numbered 0. Several cards are drawn from this deck, and their scores are calculated according to the following rule: each card numbered $k$ is worth $2^{k}$ points. If the total score of these cards is 2004, they are called a "good" card group. 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 $f(x)=\frac{1}{1+x}\left[\left(1+x^{2^{0}}\right)\left(1+x^{2^{1}}\right) \cdots\left(1+x^{2^{10}}\right]^{3}\right.$
$$
=\frac{1}{(1+x)(1-x)^{3}}\left(1-x^{2^{11}}\right)^{3}=\frac{1}{\left(1-x^{2}\right)(1-x)^{2}}\left(1-x^{2^{11}}\right)^{3},
$$

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
$$
\begin{aligned}
\frac{1}{\left(1-x^{2}\right)(1-x)^{2}}= & \frac{1}{1-x^{2}} \cdot \frac{1}{(1-x)^{2}} \\
= & \left(1+x^{2}+x^{4}+\cdots+x^{2 k}+\cdots\right)\left[1+2 x+3 x^{2}+\cdots+(2 k+1) x^{2 k}+\right. \\
& \cdots] .
\end{aligned}
$$

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 $a_{2004}=1003^{2}=1006009$.