Constructing a counting problem where the assumptions concerning repetition and order are open

Question

I don't know if this question is misplaced on this site (I'm sorry) but here goes:

I'm trying to construct a question to give to my students in high school within the subject of enumerative combinatorics. I would like to think of a question that can be interpreted in different ways - i.e. that both repetition and not repetition, and order matters and order does not matter would make sense in the situation. The intention is that the question shall lead to a generalization to the four basic counting formulas. I can only think of kind of silly situations, such as

"You have 100 songs on your computer and you have to make a playlist for a dance show with 10 songs. How many different playlists can you construct?"

In this situation both with and without repetition would kind of make sense, and the same for order. But I would just really like a better question. I intend to build a big part of my teaching of the subject on this question.

Any ideas?

Thanks!

Answer 1

EDIT: I believe I may have misunderstood the question, but since I already typed this out I'll leave it here for now. Let me know if this wasn't what you were looking for.

There is a site for math education related questions: https://matheducators.stackexchange.com/

Although it is a little uncommon so I understand that a lot of people come here anyways, which may attract more traffic.

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The examples that I like to use with my students are usually ones pertaining to the classroom, as they are a little more generally relatable. There are plenty of great examples that are more specific in application that depend on the students' interests, but I will skip them unless requested.

With and without repetition: I went to the store to buy gifts for my students. I want to distribute $n$ unique gifts to $k$ students, and I'm looking for the number of ways I can do this (without repetition). It turns out that I don't actually have one gift of each kind, but multiple of each (enough that there's enough for everyone). This automatically turns the problem into one of no repetition to one with repetition.

As an additional note: Because we usually teach the no order, repetition case using the stars and bars method, it is identical to distributing/assigning objects into boxes/groups. This does not seem like a repetition choice upon first learning about the subject, so it is an excellent way to show students how the two problems are connected and why we care about counting with repetitions.

Order matters and order doesn't matter: Select any committee within the class of $n$ students. Perhaps we need to task a team of $k$ students to clean the classroom (order doesn't matter). Now set something specific for the each of them. For instance, one student needs to put up the chairs (fun), another needs to clean the erasers (less fun), and one more needs to sweep (not fun). Now the order in which you arrange your committee matters, as they are being assigned to positions/roles $1$, $2$, and $3$.

An example with modular repetition and order: This final example can be tweaked to allow for repetition or order. Choose something that students must select independently in sequence, such as movies to watch.

• No order, repetition: How many ways can we pick $n$ movies to watch over the course of $k$ days?
• No order, no repetition: How many ways can we pick $n$ movies to watch if we don't want to watch a movie we have already seen?
• Order: Suppose we may not have enough time to get through $k$ movies, so the students should prioritize the movies they want to watch the most. Now the order matters, because there's a chance they won't get to the movies they most want to watch if they don't watch them first. The repetition idea is similar to the above.