@authenticcrane If I recall correctly, stars and bars is for both indistinguishable items. Thus your reasoning as to why it is 7 choose 2 is correct. Unfortunately, I do not think you can manipulate stars and bars for items that are distinguishable.
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Great question!! I've attached a diagram to make things clearer and I'll try to explain it here.
[Left column] first of all, as you can see, if you graph out all six solutions of (x,y), it forms this region shaped like an isosceles triangle-- in fact, it's a right isosceles triangle with side length 2 (white shaded). The solutions to this question are precisely the lattice points (points with integer coordinates) that are contained within that triangle!
But why is that?
[Right column] Well, if you think about it, we start out with a whole lattice grid of possible (x,y). That includes stuff like (x,y) = (-45840, 409574) or (x,y) = (0,0) or (x,y) = (1,2), etc etc etc. In order to narrow down this infinite # of possible (x,y) into the actual solutions, we need to consider the three constraints on (x,y):
first of all, x >= 1. That's because the problem says that x should be a positive integer, which means it can be 1, 2, 3, ....
second of all, y >=1. That's because y also has to be a positive integer.
Finally, we must have x + y <= 4 by the problem statement.
The basic idea for the algebraic solution is to graph out what each of these three constraints looks like, and "combine" them together. So if you look at the graph for x >= 1, we've shaded all points (x,y) such that x >= 1. Similarly, the pink shading represents all the points such that y >= 1. And the blue shading represents all the points such that x + y <= 4 (to do that, we first graphed the line x + y = 4, and then we shaded in everything underneath it!)
So now we have three different graphs of shaded regions, each representing different conditions. If we want ALL THREE conditions to be satisfied, all we have to do is "overlay" the graphs on top of each other and figure out what region is shaded in all three graphs! And if we do so, we see that the light yellow triangle is the region of overlap. Therefore, all the lattice points inside that triangle are the answer.
Hope this made sense to you, and feel free to reach out if you have any further questions! 😄
@spaceblastxy1428 Hi there, since this doesn't give away the solution, I'll answer here. 🙂 The question was written with the intention of not having rotational symmetry. Pretend the chairs are numbered from \(1\) to \(6,\) so that if you rotate a seating arrangement, then it's considered a new arrangement. We can clarify this in the question statement. Thanks for asking!