Question number 28. Using only one color, in how many ways can you color three different cells in this strip shown below so that no two neighboring cells are colored? And so here in this strip of seven cells, I'll make an example by using only the color orange to color three different cells so that no two neighboring cells are colored. And one easy way to accomplish that is to color the first, third, and fifth cell like that. And then I'll keep going. I'll make myself a copy here of this strip several times, five times, and keep coloring it. So I'll just redo the first example. The first, third, and fifth cells are colored. And then I'll focus here on this third colored cell and move it to three different positions. So I'll move it one over and then to the end like that. And once I have these three cells over here, I'll work on moving the first two cells. So first I'll move the middle cell over like so and then move it over once again. And by symmetry, I should be able to repeat this pattern in a mirror image. So here is another copy of my cells, and I'll have five more patterns. So here now has a mirror image of each of these. Here are these three, and then I'll have something like this over here. And there we go. Those are mirror images of the patterns, and that accounts for all possibilities. We cannot make any more moves. If we try to move any of the cells over, we'll just create a pattern that already exists. For example, if I take this red cell over here and try to shift it in either direction, I'll just repeat the pattern that's above, and same thing over here. If I shift any of these cells over in any direction, or maybe even two at a time, I'll just create a pattern that already exists. So these are all the possibilities, and we can count that there are, in fact, ten different ones. So ten is here the answer to question 28. That is choice E.

coloring cells

non-neighboring

unique arrangements

combinatorial problem

symmetry