Can you please hint on this? at least the topic of the graduate level course? I am curious.....thanks
Frankly, I have not taken graduate level combinatorics myself. My gut sense is that the majority of highly complicated problems would be solved via technology. Take, for example, the freakish problem you proposed --- 100 tiles, 8 colors, and let's say 20 tiles of each color. How many possible combinations?
By paper and pencil methods, that could take a very long time. I think the shortest way for anyone to do this would be to write a program that enumerates possibilities, somehow counting only the ones that fit the constraints. I'm not really a programmer, but I believe this would be a not-very-hard thing to program.
As I said, all this is well out of the league of anything the GMAT would expect you to do. Notice, for example, in the original problem in this thread, we were given the saving grace that there were five
of each color --- just one less than the number of tiles on the floor
--- and that allowed for the extremely slick and elegant solution discussed above. The GMAT is absolutely not interested in posing problems that require nightmarish levels of analysis to solve. By contrast, they are all about problems that lend themselves beautifully to surpassingly elegant solutions. Most larger combinatorics problems do NOT fall in that category. That's why those question are irrelevant to GMAT preparedness.
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