I’m currently working on a finger exercise for mit6.00.2x, a MOOC in computaional thinking, and was having some issues.
First of all: don’t worry, I don’t need you to do my homework for me, I just want to clarify an aspect of the question.
So, the original question asked us to program the computer to find all subsets of a list of items to go into a bag. It said that "The number of possible combinations to put n items into one bag is $2^n$."
So far, so good.
However, the extension of the question is to program the computer to output the amount of combinations possible to put n items into 2 bags: an item must be in either bag or neither.
Now we arrive at my question: At this point it is said that there are $3^n$ possible combinations for this, as each item has three possibilities instead of two.
This makes sense, but I actually did the problem with two items and only found seven combinations:
[][1], 2 outside
[1][], 2 outside
[][], 1 and 2 outside
[][1,2]
[2][1]
[1,2][]
[1][2]
So, there are 2 items, there should be $3^2=9$ combinations, but there are only 7.
I'm sure I'm missing something....help!