Quandaries
and Queries 

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. 

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: 53 = 15, means than if you have five ways to make a first choice, and then 3 ways to make a second choice, then there are fifteen possible outcomes of the two choices. The fact that you got the right answer by counting all possibilities shows that your architectural mind understands this fact enough to list all possibilities correctly. All you need to do now is to relate this counting procedure to the meaning of multiplication. 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 3D. 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 ! 

