Final answer:
The distribution of dollar bills to discrete math students can be solved as a combinatorial problem using stars and bars. With at least one dollar per student, use (16 choose 4) for 17 dollars and (15 choose 4) for 16 dollars. When allowing students to receive nothing or adding maximum caps, the complexity increases and may require advanced methods.
Step-by-step explanation:
The distribution of 17 one-dollar bills to 5 discrete math students can be calculated using combinatorial methods. When each student must get at least one dollar, we actually need to distribute 12 dollars since 5 are already given (one to each). This is a stars and bars problem and the solution is choosing 4 partition points out of the 16 available spots (17 dollars - 1 for each student), which can be denoted as (16 choose 4). For scenario (b), where students can get nothing, we distribute 17 dollars without restriction into 5 categories, which is (21 choose 4). Scenario (c) places a cap on how much each student can get, it becomes a restricted stars and bars problem and will require more detailed calculations using generating functions or dynamic programming.
Similar approaches are used for the distribution of 16 one-dollar bills to the same students. The main difference is the initial number of dollars before considering the at least one dollar requirement, making the problem (15 choose 4) for part (a) because we'll have 11 dollars to actually distribute after giving one to each student.