Finding the number of arrangements where the order is constrained and then dividing into groups

Question

Hello everybody! The above problem you see is a problem I could not solve. :(

Basically this problem, I guess, is asking us to arrange the courses into the order wanted and dividing the courses into the minimum number of semesters. But I am having a lot of difficulty apparently on how to arrange them and find the minimum number of semesters we want. I tried going one by one, step by step to solve this problem but that doesn't work especially for the longer parts. :( I think we'll have to think of an algorithmic approach to this problem but how? I know the answers and they are: $7,6,5,8$..

BTW I have asked some more problems. Can anybody please help me with them?

Finding the number of ways in which columns can be connected such that even number of specific columns are selected.

Arranging objects where some $depend$ upon some other(s)

Finding the subsets of people sitting around a circular tables

Minimizing the number of operations of subtracting one so as to get to $0$

Any help would be appreciated. Thanks!

Answer 1

You can solve these via topological sorting.