Is there a notation for the act of choosing k items from a set of n items? Similar to the way C(n, k) represents the number of ways to choose k items from n items, is there a compact notation to represent those specific choices? Like this:
Choose({a, b, c, d}, 3) = {{b, c, d}, {a, c, d}, {a, b, d}, {a, b, c}}
 A: If $k$-element subsets of various sets $A$ play a rôle in your text you can introduce the notation $${A\choose k}$$ for the set of these subsets at the beginning of your argument.
A: If your original set $A$  has $n$ elements and you are looking for subsets $ B$ of $m$ elements , then  you have $$ S= \{ B\subseteq A: \text {   }|B|=m\}$$ where $|B|$ stands for the cardinality of $B$
A: You can use the elementary symmetric polynomial.
For your example we have
$$e_3(x_a,x_b,x_c,x_d)=x_ax_bx_c+x_ax_bx_d+x_ax_cx_d+x_bx_cx_d$$
where, for example, $x_ax_bx_c$ represents the subset $\{a,b,c\}$.
In general if set elements are $\{1,\ldots,n\}$ then $e_k(x_1,\ldots,x_n)$ is easily seen to be given by the $z^k$ coefficient of $\prod_{k=1}^{n}(1+zx_k)$ i.e.
$$\prod_{k=1}^{n}(1+zx_k)=\sum_{k=0}^{n}e_kz^k$$
so, we may use the "evaluate $z^k$ coefficient operator": $[z^k]$ to give your shorthand for $e_k$:

$$e_k(x_1,\ldots,x_n)=[z^k]\prod_{k=1}^{n}(1+zx_k)\tag{1}$$

In your case
$$e_3(x_a,x_b,x_c,x_d)=[z^3](1+zx_a)(1+zx_b)(1+zx_c)(1+zx_d)$$
So I would suggest you could use either side of $(1)$. 
It's, comparatively, not that compact for small sets but you can see that $(1)$ is the same for arbitrary set size: $n$.
