Example 1 Allocate 24 volunteer slots to 3 schools. Then the number of allocation methods where each school gets at least one slot and the number of slots for each school is different is $\qquad$ kinds. ${ }^{[2]}$
Let the quotas allocated to schools A, B, and C be $x$, $y$, and $z$ respectively.
First, without considering that $x, y, z$ are pairwise distinct.
From $x+y+z=24$, we get a total number of combinations $\mathrm{C}_{23}^{2}$.
Next, we separate out the number of positive integer solutions $(x, y, z)$ that are not pairwise distinct.
From $2a+b=24$, we know $b=2c$.
Therefore, $a+c=12$, which has 11 solutions.
Among these, one solution satisfies $a=b=8$, corresponding to one triplet $(8,8,8)$;

Additionally, 10 solutions $(a, a, b) (a \neq b)$ correspond to $10 \mathrm{C}_{3}^{1}$ positive integer triplets $(x, y, z)$.

Thus, the number of pairwise distinct positive integer triplets $(x, y, z)$ is
$$
\mathrm{C}_{23}^{2}-1-10 \mathrm{C}_{3}^{1}=222 .
$$