#combinatoric problem

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.