1) Three identical light beams are to be installed in the vertices of a hexagonal tower, ie., hexagonal cylinder, with 12 vertices.
a) No other restriction applies?
b) Only the top vertices are to be used?
c) Only one beam goes at the top?
2) There are ten identical books, and one each of 10 different books. How many groups of ten books are there?
3) A certain number of shampoo containers are placed in groups of 12, and 8 remain. But, placed in boxes that hold 35, only 2 remain. Find all the possible numbers of containers. In particular, what is the smallest possible number?
a) Based on b, if three lights are on the same surface, there are 20*2=40 ways; Based on c, if three lights are on the different surfaces, there are 90*2=180 ways. Therefore, the total ways: 40+180=220 ways.
b) If only the top vertices are to be used, there are two scenarios: All three are next to each other and there are 6 ways; If only two of them are next to each other, there are 12 (6*2)ways: If none of them are next to each other, there are only 2 ways; The ...
This solution provides steps to solve each of the discrete math calculations.