# Partitioning objects in combinatorics

When you come to explain dividing given n number of objects into k number of groups, is it good to describe the cases involved using an example to cover as many cases as possible in order to give better opportunity to students to understand the difference by comparing ?
What would you think about the following,
Let consider dividing 5 objects into 3 groups;

1. putting 5 similar balls into 3 similar baskets.
2. giving 5 similar balls to 3 persons.
3. putting 5 different balls into 3 similar baskets.
4. giving 5 different balls to 3 persons.
Are these cases enough to give full knowledge about partitioning with an extra restriction such as each of the partition should have at least one item?
Do we need to add another 7 objects of a different kind to the first two cases in latter stages to make students familiar to much complicated situations?
Do you have much better way of introducing this to students?

When I was introduced to combinatorics, we were concentrating on various counting techniques rather than on various types of situations in which the counting has to be performed with the idea that the techniques are not too many but the number of setups one can invent is immense. Interestingly enough, ArtOfProblemSolving mainly takes the same approach, so if you look at their videos, you'll see counting trees, independent choice, mutually exclusive choice, complementary counting, Venn diagrams, principle of inclusion-exclusion, compensating for overcounting using symmetry, counting with restrictions, and so on, and so forth rather than a discussion of particular setups in some order.

However the argument about what is primary in mathematics: an object or a method is, probably, older than all of us. I'm in the "method" camp, so I try to go one technique a time when teaching some introductory level classes like that with various seemingly unrelated setups where the same technique or set of techniques is used. But if you want to go "object by object" as the classification in your post suggests, that is fine too if you think in advance of what setups to choose (arrangements, partitions, assignments, etc.) and how to organize the stuff in each category so that not to overload students' memory because many of them just tend to memorize examples rather than absorb the underlying ideas at this level.

There seems to be no way that is unconditionally better (or worse). Just think of what skills you want your students to acquire in the end. Ideally, they should know some amount of techniques, some set of classical setups, the standard matching between them, and, when going into the uncharted territory, be able to reduce the new setups to the old ones, or match them with appropriate (combinations of) known techniques directly, or add some twists to the existing methods to account for the peculiarities of the situations. However in real life one can never go that far in the short introductory course, so one has to sacrifice this and that, cut corners here and there, and reduce stuff to the simplest cases now and then. The choice of this compromise is by no means unique and depends on both the teacher preferences and the student abilities.

So my final answer to your direct question is "Start with what seems most logical to you and adjust the level and content of the lessons to the level of the students within reasonable limits. For the latter I usually set up some minimal requirements in the beginning (for myself, not for telling the students) and if I see that I cannot successfully do something above them, I consider my lectures too hard for this particular audience, but if I have trouble doing something within them, I consider the students too weak or not ready for the course. Normally those minimal requirements are more or less aligned with the standard description of the course but, of course, you always have some power of interpretation here.

Just my two cents.

• Thank you very much sharing your suggestions, I would greatly appreciate,if you can share few other references like ArtOfProblemSolving or text books related to this issue. May 25, 2023 at 1:36
• You can google art of problem solving. Here: artofproblemsolving.com May 25, 2023 at 2:20
• @SueVanHattum thanks for sharing the link, but I requested other than that in my comment. May 25, 2023 at 7:44
• @JanakaRodrigo The AoPS videos devoted specifically to counting and combinatorics can be found at artofproblemsolving.com/videos/counting They are all free, so you are welcome to watch and to recommend them to your students if you think they are helpful. As for textbooks, unfortunately, I'm not sufficiently familiar with literature in English to recommend anything (though AoPS has its own books and, perhaps, some references too; just bear in mind that they are somewhat geared towards "Olympiad type mathematics" in general). May 26, 2023 at 20:04