Determine the number of ordered pairs (A,B), where A is a subset of B and B is a subset of {1,2,....,n}.
For a particular subset and B has k elements then there are subsets of B. Hence there are ordered pairs (A,B) where . BUT that is only for one particular subset.
Let’s do the general case: for any number j, , there are subsets of having j elements. Thus there are pairs of the form (A,B) where and |B|=j.
The total count: .
Comparing that with the binomial formula , you see that the total count is (put x=2, y=1).
You can see this answer directly from the original question if you argue as follows. For each integer from 1 to n, there are three possibilities: (a) it belongs to both A and B, (b) it belongs to B but not A, (c) it belongs to neither. Total number of possible choices: (but I wouldn't have thought of that if I hadn't seen Plato's answer first).