We have a child's toy (FP Stack and Roll Ball Set). There are 10 half dome cups that can be stacked in a variety of ways. It was hotly debated among our group how many ways to stack them are available. 10! seems too easy. The question arose at a Christmas party with several engineers playing with the toy and debating about the number of permutations that are possible. I said there were likely millions of combinations and that was doubted and a thousand was the more heavily agreed number. Link to the toy image for more clarification: http://mommyftw.com/wp-content/uploads/2010/03/cups-300x300.jpg
More Details: - Each dome can stack with all domes smaller than itself (think stacking tower). - Each dome can snap into a ball shape with each dome one size larger and one size smaller than itself. - When two domes are snapped into a ball shape other smaller domes can be snapped onto the outside of the ball. - Two smaller domes snapped into a ball shape can fit within two domes snapped into a ball that are larger. - Order of stacking doesn't matter as long as the stacked domes are increasingly smaller.
How would you calculate the number of permutations? What is the total number of permutations? I'm looking for the largest number of permutations possible and it's been 10-12 years since I've been in college.