Problem: Shades of Pre-registration

Suppose that you are building a schedule from among \(k\) distinct possible courses. Give combinatorial descriptions of each of the cardinalities described below. For each description, check your work by instantiating a concrete set of courses, and demonstrating that your description works for each such concrete example.

1. The number of five course schedules (i.e., sequences of courses) you can build from these courses.

2. The number of sets of three courses (i.e., unordered collections) you can build from these courses.

3. The number of four course schedules that contain both Basket Weaving and Underwater Knitting.

   (Hint: choose positions for Basket Weaving and Underwater Knitting. This implies where the remaining two courses will go. Finally choose which courses go into those remaining two slots.)

4. The number of four course schedules that do not contain both Basket Weaving and Underwater Knitting.

   (Hint: leverage your answer to the previous part!)

5. The number of six course schedules that include Calculus I and Computer Science I but do not place Calculus I after Computer Science I in the schedule.

   (Hint: similarly to the previous part, choose positions for the two named courses. But here, you need to remove the possibilities that get the order these courses swapped.)

6. (Bonus Problem) The number of six course schedules that include Calculus I and Computer Science I but do not place Calculus I immediately after Computer Science I in the schedule.

   (Hint: unlike the previous problem, we only want to ensure that Calc I does not appear in the next time slot after CS I. How can we break up the possible positions of the two courses to get a handle on these situations? How do we then account for overcounting?)