First of all, what you are talking about is called a set. Second, it is correct that the number of distinct sub-sets that can be generated out of a set is equal to 2^m where m is the number of elements in that set. We can arrive at this result if we take an example of 3 elements:
S = {a, b, c}
Now to generate every sub-set we can model the presence of an element using a binary digit:
xxx where x is either 0 or 1
Now lets enumerate all possibilities:
000 // empty sub-set
001
010
011
100
101
110
111 // the original set it self!
Lets take 011 as an example. The first digit is 0 then, a is not in this subset, but b and c do exist because their respective binary digits are 1’s. Now, given m(e.g 3 in the above example) binary digits, how many binary numbers(sub-sets) can be generated? You should answer this question by now 😉