So, in my secondary Algebra II class we have gotten to the unit on probability and counting and as part of that we are looking at permutations. We have looked at permutations of $n$ objects, permutations of $n$ objects taking $r$ at a time, and permutations with identical objects. The first two types of permutations my students understand pretty well, both intuitively and the formulas. We talked about these in terms of "slots and options", i.e. "If you have three slots and ten objects, then you have 10 options for the first slot, 9 options for the second slot, and 8 options for the third slot and by the fundamental counting principle we multiply these to get the total number of permutations." I was actually able to have the students get up and physically perform these permutations by standing in different orders in front of the class and this really seemed to cement in these first two types of permutations.
The problem arose when we got to the idea of permutations with identical objects. I tried to explain it as "the total number of permutations for the objects ($n!$) and then divide out the number of repeats ($r_1!r_2!...$)" A few of my students were good with this but for the most part, the class was not able to see the connection between the situations we were looking at and the formula and ended up very confused. Does anyone have a better, more intuitive way to explain permutations with identical objects? or perhaps a demonstration, similar to what I did with having my students line up in front of the class, that could make this concept clear?