The approach to this problem is very simple.
Directly using the AM-GM inequality, we get
$$
\begin{array}{l}
\sum_{i=1}^{4} a_{i} b_{i} c_{i} \geqslant 4 \sqrt[4]{\prod_{i=1}^{4} a_{i} b_{i} c_{i}}=4 \sqrt[4]{24^{3}}>43 . \\
\text { Therefore, } \sum_{i=1}^{4} a_{i} b_{i} c_{i} \geqslant 44 .
\end{array}
$$

The equality can be achieved:
$$
1 \times 3 \times 4+2 \times 2 \times 3+3 \times 4 \times 1+4 \times 1 \times 2=44 \text {. }
$$

Thus, the minimum value sought is 44.
Question 1 Let $a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}, c_{1}, c_{2},$ $c_{3}, c_{4}$ all be permutations of $1,2,3,4$. Find the minimum value of $\sum_{i=1}^{4} a_{i} b_{i} c_{i}$.