4. (40 points) A rectangular box with dimensions $a_{1} \times b_{1} \times c_{1}$ can fit into another rectangular box with dimensions $a_{2} \times b_{2} \times c_{2}$ if and only if $a_{1} \leqslant a_{2} 、 b_{1} \leqslant b_{2} 、 c_{1} \leqslant c_{2}$. Therefore, among the rectangular boxes with dimensions $a \times b \times c$ (where $a 、 b 、 c$ are integers and $1 \leqslant a \leqslant b \leqslant c \leqslant 5$), what is the maximum number of boxes that can be selected such that no box can fit into another?
4. Among the boxes that meet the conditions, there are 5 types of boxes that are cubes, and 20 types of boxes that are square-based but not cubic prisms, because they have five different heights and four different choices for the square side length.

In addition, there are 10 types of boxes where the length, width, and height are all different, because three can be chosen from five lengths, resulting in $\frac{5 \times 4 \times 3}{3 \times 2}$ types. Table 2 categorizes these 35 types of boxes based on their length + width + height values. Within the same category, boxes with smaller length + width + height values can fit into boxes with larger length + width + height values.
Table 2
\begin{tabular}{|c|c|c|c|c|c|}
\hline Length+Width+Height & First Class Second Class Third Class Fourth Class Fifth Class \\
\hline 15 & 555 & & & & \\
\hline 14 & 455 & & & & \\
\hline 13 & 355 & 445 & & & \\
\hline 12 & 255 & 345 & & 444 & \\
\hline 11 & 155 & 245 & 335 & 344 & \\
\hline 10 & 145 & 244 & 235 & 334 & \\
\hline 9 & 135 & 144 & 225 & 234 & 333 \\
\hline 8 & & 134 & 125 & 224 & 233 \\
\hline 7 & & 124 & 115 & 223 & 133 \\
\hline 6 & & & 114 & 222 & 123 \\
\hline 5 & & & 113 & & 122 \\
\hline 4 & & & & & 112 \\
\hline 3 & & & & & 111 \\
\hline
\end{tabular}

One box with a length + width + height value of 9 can be selected from each category, i.e., $(1,3,5)$, $(1,4,4)$, $(2,2,5)$, $(2,3,4)$, and $(3,3,3)$. Each of these boxes has at least one dimension that is larger than the others, making it impossible for them to fit inside each other. Therefore, the maximum number of boxes that can be selected is 5.

Note: Giving an example with 4 boxes earns 10 points, giving an example with 5 boxes earns 20 points. Simply stating 5 boxes without giving an example earns 0 points. Proving the maximum of 5 boxes earns 20 points.