It's a common theme in mathematics that, if there's no canonical choice (of basis, for example), then we shouldn't make a choice at all. This helps us focus on the heart of the matter without giving ourselves arbitrary stuff to drag around.
However, in this question, I'm looking for examples of problems solved by a specific type of "not making a choice" – namely, making all available choices, and looking at all the end results together as a whole. We can't necessarily discern any individual piece, but the average behavior, or some other information about the big picture, provides (or at least points towards) a solution.
I really wish I had an example of this phenomenon to provide, but even one escapes me at the moment, which is what spurred me to ask this question. I imagine combinatorics is full of examples; unfortunately I haven't really studied that field in any depth yet.
Something close to what I'm after is Burnside's (a.k.a. not-Burnside's) Lemma. There's no good way of directly counting orbits, i.e. choosing to look at a particular orbit one at a time, so we just look at the average number of fixed points of elements of $G$ (I'm reluctant to call this an example of the kind of result I'm looking for because I'm not entirely clear on why the fixed points of an element should be thought of as substitutes for orbits. Perhaps that's a separate question).
This question is in a similar vein.
Best Answer
Another classical example of "looking at all choices instead of one" is the idea of the fundamental groupoid of a topological space. Instead of choosing one base point and letting all loops begin and end in this point one considers all paths between all points (modulo homotopy). This notion makes theorems like the Seifert-Van-Kampen-theorem much more natural. One does no longer have to add technical conditions that certain intersections contain the base point and are path connected.