【Example 5】 Now we have 6 Chinese chess pieces, including one red and one blue for each of the types: Chariot (車), Horse (馬), and Cannon (炮). We need to arrange them in a row such that the red Chariot is to the left of the blue Chariot, the red Horse is to the left of the blue Horse, and the red Cannon is to the left of the blue Cannon, but they do not necessarily have to be adjacent. How many different arrangements are there?
Since the relative order of the two chariots, two horses, and two cannons is fixed, we only need to consider which two positions will be used for the chariots, horses, and cannons, respectively. Therefore, it ultimately boils down to a permutation problem of elements that are not entirely distinct. Hence, we have
$$
\left(\begin{array}{ccc}
6 & \\
2 & 2 & 2
\end{array}\right)=6!\cdot(2!)^{-3}=90
$$

different arrangements.