Course Schedule III - Solution & Explanation

This approach involves using a greedy method combined with a max-heap (or priority queue) to choose which courses to take.

1. Sort the courses based on their last day to prioritize courses that need to be completed earlier.
2. Iteratively add each course to a max-heap which allows keeping track of the courses currently selected based on their durations.
3. If adding a new course exceeds the current day deadline in the schedule, remove the course with the longest duration from the heap. This is done because replacing the longest course is less detrimental to the schedule than other courses.
4. The size of the max-heap at the end represents the maximum number of courses that can be fit into the schedule.

This Python solution sorts the courses based on their last available day. It utilizes a max-heap (by pushing negative durations) to manage the courses being taken such that the longest duration course is easily removable when a constraint breach occurs. The heap size at the end represents the number of courses taken.

This approach defines a DP table where `dp[i][t]` represents the maximum number of courses we can fit into our schedule given the first `i` courses considered and `t` days available. The decision at each step involves either taking or not taking a course. The transition function will decide by taking the maximum between taking the current course (if it fits) or not taking it.

This Java solution uses a dynamic programming approach to solve the problem efficiently by iterating backwards through possible days to ensure overlapping updates do not occur, thus propagating maximum course counts correctly at each considered day.