For example, let's say I wanted to denote any arbitrary, $2$ number combination of the letters, A, B and C. So you can have AB, AC, and BC. Say you wanted a way to represent any given combination, is there a shorthand way to denote this idea?
The reason why you may want to do this, is that whilst every combination is unique, every combination may share a unique property, which wouldn't be seen given other combination sets (i.e combinations derived from other pairs of value, say, $\{C,D, E\}$, or $\{1, 2, 3\}$ etc.) if you wanted to refer to a certain property that every combination posses (relative to a specific set of values), a shorthand for this would I think be convenient :)
And of course, the same can be said for permutations.
As a spinoff of that notation that $n \choose k$ denotes the number of $k$-element subsets of a set of size $n$, we can define $S \choose k$ to denote the set of all $k$-element subsets of a set $S$. So, to say that you're thinking about one of the sets $\{A,B\},\, \{B,C\}$ or $\{A, C\}$, you might write something like "$\Delta \in {{\{A, B, C\}}\choose{2}}$" to signify that the set $\Delta$ is one of the above $2$-element subsets of $\{A, B, C\}$.
The notation is reasonably "natural" in the sense that $$\left\lvert {S \choose k }\right\rvert = {{\lvert S \rvert} \choose {k}}.$$
I personally like this notation, and know it is used to some extent, but I have no idea how popular it is "in the field." As for permutations, I have never seen anything comparable.
And, as a general rule, it never hurts to make a note about what your notation means, if you're not just writing things down for yourself (and honestly, it couldn't hurt then, either!)