Three. (50 points)
There are 12 rabbit holes, and holes $1,2,3,4,5,6,7,8,9$ contain a total of 32 rabbits. Holes $4,5,6,7,8,9,10,11,12$ contain a total of 28 rabbits, holes $1,2,3,7,8,9,10,11,12$ contain a total of 34 rabbits, and holes $1,2,3,4,5,6,10,11,12$ contain a total of 29 rabbits. How many different possible distributions of the number of rabbits in each hole are there?
Let there be $x_{2}$ rabbits in the $i$-th hole $(1 \leqslant i \leqslant 12)$, and let $y_{j}=x_{3,2}+x_{3,-1}+x_{3 j}(1 \leqslant j \leqslant 4)$. 
From the given information, we have $\left\{\begin{array}{l}y_{1}+y_{2}+y_{3}=32 \\ y_{2}+y_{3}+y_{4}=28 \\ y_{1}+y_{3}+y_{4}=34 \\ y_{1}+y_{2}+y_{4}=29\end{array}\right.$
Solving this, we get $y_{1}=13 \quad y_{2}=7 \quad y_{3}=12 \quad y_{4}=9$
On the other hand, the equation $x_{3,-2}+x_{3,-1}+x_{3 j}=y$ has $C_{y+2}^{2}$ different non-negative integer solutions. Therefore, there are $\prod_{t=1}^{4}\left[C_{y_{t}+2}^{2}=18918900\right]$ in total.