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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.

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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!)

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    $\begingroup$ For what it's worth, I use this notation and can reasonably be considered to be "in the field". $\endgroup$ Jul 14 '16 at 0:58
  • $\begingroup$ @AustinMohr Good to know! I had assumed as much, but for me, I rarely encounter anyone needing to specify that the subsets are a certain size. So I see $2^S$ plenty often, but never anyone needing $S \choose k$. $\endgroup$
    – pjs36
    Jul 14 '16 at 1:02
  • $\begingroup$ Cheers pjs36, the only thing I don't get is the | | symbol, at least in this context. Oh, and is subset here the same as some combination? $\endgroup$ Jul 14 '16 at 1:03
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    $\begingroup$ @user2901512 The notation $|S|$ just means the cardinality, or size, of the set $S$. So $|\{A,B,C\}| = 3 = |\{C, D, E\}|$, for example. $\endgroup$
    – pjs36
    Jul 14 '16 at 1:05
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    $\begingroup$ Oh sweet, I never knew/thought of that, that's so cool ^.^ $\endgroup$ Jul 14 '16 at 1:07

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