A problem from research:
So there are a set of boxes you want to get that are numbered 1 to 5 in the figure above. There are different methods of getting some of these boxes that are labeled A through D in the figure above. Method A will get boxes 1, 2 and 5. So the game is to find the minimum number of methods necessary to get all the boxes. The example above requires 2 methods: either A & B or C & D. Not so bad. At the very worse, you could try all the possibilities:
1 method (4 choices): A, B, C, D
2 methods (6 choices): A&B, A&C, A&D, B&C, B&D, C&D
3 methods (4 choices): A&B&C, A&B&D, A&C&D, B&C&D
4 methods (1 choice): A&B&C&D
15 possibilities = 2^4 -1. You can see that this is the maximum number because you can choose to either use a method or not -- meaning that there are 2 choices for each method and the one where no methods are used is boring.
My real problem had 8064 methods and 6411 boxes. This means that there are 2^8064 possibilities (or 10^2400 = (1 Googol)^24 )