5.085 Six boxes of different materials are delivered to eight floors of a construction site. In how many ways can the materials be distributed across the floors? In how many of these ways will at least two materials be delivered to the eighth floor?
Solution. For each box, there are 8 ways to distribute it across the floors. Therefore, for six boxes, there are a total of $8^{6}$ such ways.

If the 8th floor does not have any boxes, then for each box, there are only 7 ways to distribute it across the seven floors, i.e., a total of $7^{6}$ ways for 6 boxes. If there is exactly one box on the 8th floor, then the other 5 boxes can be distributed across the remaining floors in $7^{5}$ ways. The number of ways to choose the box for the 8th floor is 6. Therefore, the number of ways in which there is exactly one box on the 8th floor is $6 \times 7^{5}$.

Thus, the total number of ways in which there are no more than one box on the 8th floor is $7^{6} + 6 \cdot 7^{5} = 13 \cdot 7^{5}$. From this, it follows that the number of ways in which there are at least two boxes on the 8th floor is $8^{6} - 13 \cdot 7^{5}$. Answer: $8^{6} ; 8^{6} - 13 \cdot 7^{5}$.