Consider the equation $${n\choose0}+{n\choose1}+{n\choose2}+\cdots+{n\choose n}=2^n.$$
By definition, every term in this equation represents a count of subsets:
- $n \choose k$ represents the number of $k$-element subsets of an $n$-element set.
- $2^n$ represents the total number of subsets of an $n$-element set.
I realize that, at this point, the fact that the equation holds for any $n\ge0$ is self-evident (as we are counting the same thing in two different ways). I was wondering if the following (admittedly clunky) line of thought was also valid:
If we were to replace the counts of the subsets with the subsets themselves (treating the addition as a disjoint union), we would end up with a set of all subsets of an $n$-element set on both sides.
The idea of a "set of sets" is new to me, and having it appear in this supposedly simple line of thought brings some doubts to my mind. Is the reasoning valid?
Yes, it's fine. There's a set of all subsets of a given set, and it decomposes as a disjoint union of the sets of subsets of each possible size. This gives what is called a bijective proof of the original identity; for much more on this see Art Benjamin's Proofs that Really Count.