Hi James, first off, the main point of the homework is to practice and review, so I wouldn't worry too much about your score on the homework. The main goal is to keep improving.

I think the difficulty you're having is what makes a lot of counting questions difficult. It is very important to think clearly and pay attention to details and assumptions in the question.

One thing I definitely recommend when reviewing problems is understanding why you got a question wrong by changing the wording of the question to make your answer correct to clearly understand the assumptions for the problem.

Let's look at one of the examples from last week's class: "A train with 20 passengers makes 7 stops. How many ways are there for the passengers to get off the train?"

• The intended answer is $7^{20}$. What we're really assuming here is that we just need to know where each (distinct) passenger gets off the train, and any number of passengers can depart the train at each stop. Thus, a fully detailed version of the question might be something like "A train with 20 passengers makes 7 stops. Some, none, or all passengers get off at each stop, and the order they depart the train does not matter. How many ways are there for the passengers to get off the train?"
• Now suppose you mistakenly think the answer is $\displaystyle \binom{20+7-1}{20}$ using stars and bars. (This was part b of the example.) In this case you're making many of the same assumptions as above, but now treating each passenger as identical. But are people really identical? Probably not!

I know this is a fairly simple example, but I do think this strategy can help. When reviewing the problems, understand how the wording would need to change to fit the answer you gave, so you start noticing where mistakes are being made. This can be time consuming at first, but will be worth it in the long run.