Example 25 (2004 National Women's Mathematical Olympiad Problem) A deck of tri-color cards, totaling 32 cards, includes 10 cards of each of the red, yellow, and blue colors, numbered $1, 2, \cdots, 10$; there are also a big joker and a small joker, each numbered 0. From this deck, several cards are drawn, and then the score is calculated according to the following rule: each card numbered $k$ is counted as $2^{k}$ points; if their total score is 2004, then these cards are called a "good card group." Try to find the number of "good card groups."