We have studied the standard way of ascertaining the total number of subsets of a set by using the concept of combinations ( or binomial coeffecients ). I came across an alternate derivation for this fact. The derivation goes as follows:-
Consider the problem of placing the $r$ elements of $A$ in two boxes. Corresponding to each placement, we can define a subset of $A$ by taking the elements placed in box 1 and discarding the elements placed in box 2. Since there are $2^r$ ways to place the $r$ elements, there are $2^r$ subsets of $A$.
My doubt is that, although I understand the line of logic behind this reasoning, but I think that this reason can also be given with 3,4,5 ( and so on ) number of boxes. All we have to do is to observe only one box and discard all the other. Where am I wrong?
Reference:- Elements of Discrete Mathematics, Liu ( page 70, example 2.6 ).