Example 8 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 the score is 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.
Let $a_{n}$ denote the number of card groups with a total score of $n$, then $a_{n}$ is equal to the coefficient of $x^{n}$ in the expansion of the function $f(x)=\left(1+x^{2^{0}}\right)^{2}\left(1+x^{2^{2}}\right)^{3} \cdots$ $\left(1+x^{2^{n}}\right)^{3}$. $n \in\{1,2, \cdots, 2004\}$.
$$
\begin{aligned}
f(x) & =\frac{1}{1+x}\left\{\left(1+x^{2^{n}}\right)\left(1+x^{2^{2}}\right) \cdots\left(1+x^{2^{n}}\right)\right\}^{3}=\frac{1}{(1+x)(1-x)^{3}}\left(1-x^{2^{\prime \prime}}\right)^{3} \\
& =\frac{1}{\left(1-x^{2}\right)(1-x)^{2}}\left(1-x^{2^{n}}\right)^{3}
\end{aligned}
$$

Notice that $n \leqslant 2004<2^{11}$, so $a_{n}$ is equal to the coefficient of $x^{n}$ in the expansion of $\frac{1}{\left(1-x^{2}\right)(1-x)^{2}}$.
And $\frac{1}{\left(1-x^{2}\right)(1-x)^{2}}=\left(1+x^{2}+x^{4}+\cdots+x^{2 k}+\cdots\right)\left(1+2 x+3 x^{2}+\cdots+(2 k+1) x^{2 k}+\cdots\right)$,
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 groups is $a_{2004}=1003^{2}=1006009$.