Quandaries
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My name is Jose I'm an architect student and have a question on combinations. I have a grid of 3 x 3, hence a total of 9 spaces. I have 3 elements to place in this grid. How many possible ways are there of arranging this elements on this grid ? (order, orientation not important) First putting the elements each in its own space and secondly allowing the elements at a given moment to "share" one space. Since I got kind of obsessed with this I went ahead and graphically did all the combinations allowing "sharing", a grand total of 729. How could I have known this before hand ? Thank you very much. |
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Hi Jose, If you look at 729 you'll notice it is 9x9x9 -- there are 9 spots for the 1st element to go in, 9 for the 2nd and 9 for the 3rd. Actually this is exactly what multiplication "does'' in a sense:
5 Penny and Claude Jose wrote back: As side facts let me share the following with you: Without an organized system to develop the possible combinations ( I mean when I started drawing them on Tuesday ) it took me about 5 hours to come up with around 200 combinations, completely aware of so many more but without order to develop them I went to bed. On Wednesday, it "clicked", I visualized a method, and I was able to develop all of them, the whole 729 in about 3 hours, I was so proud of myself ! Specially after your confirmation. From your input I can deduct that if I don't allow sharing the formula would be 9x8x7 and hence 504 the possible combinations in which every element is in a single space. As an architect the next step, is ? .... of course, to go 3-D. So my 3x3 grid becomes a cube with 27 spaces, still the same 3 elements. Allowing sharing that would mean 27x26x25 = 17550! Drawing this ones out is really a task beyond my patience, maybe when I retire ! Although I must confess, it is a very attractive idea. I mean architecturally speaking, the possible combinations in space, with these few variables, anyway, I'll let you guys know. The ultimate purpose of this exercise comes from a small house I'm designing. I wanted to find the way 3 different areas ( in other words rooms) each with a unique function, i.e. bathroom, kitchen and sleeping spaces, could be arranged. That's why I wasn't interested in the combinations in which they shared a space, rather uncomfortable to sleep in the table and bath in the sink ! |
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