How to formularize the total number of subsets of a set where some elements may be chosen from a finite list? Consider the following example:
The set $A = \{1,\{a,b,c,d\},\{e,f\},5,7,11,13,17,19\}$, where the second and third elements represent the sets of choices for the second and third elements. We could write, $A = \{1,\{|4|\},\{|2|\},5,7,11,13,17,19\}$ 
Then the total number of subsets for $A$ would be $2^9 + 4*2^8 + 3*2^7$. I found this by writing out the different choices for $A$ and considering which subsets would have already been counted after the initial $2^n$, where $n = |A|$:
# subsets for $\{1,a,e,5,7,11,13,17,19\} = 2^9$
$+$# subsets for $\{1,b,e,5,7,11,13,17,19\} = 2^8$
...
Can and how may this be formularized for an arbitrary number and size of "choice-type" elements in $A$?
 A: I don't really know what you mean by "formalize". Your definition is a little muddled, so I suppose stating it in more precise language could be helpful.
Here's how I would express what you're trying to say:


*

*A menu is a family of sets.


You may think that not every element of a menu has to be a set, some of the elements might be numbers. For instance, you might have something like $\{1, 2, \{a ,b\}\}$. But this is equivalent to $\{\{1\}, \{2\}, \{a ,b\}\}$.


*

*If $(A_i)_{i\in I}$ is a menu and $J\subseteq I$, then let $\phi:J\to \bigcup_{j\in J} A_j$ be a function such that $(\forall j\in J)\ \phi(j)\in A_j$. Then the image set $\phi(J)$ of $\phi$ is a selection from A.


The second point basically says that a selection from $A$ is a set that contains at most one element from each set in $A$. The reason we can't just say exactly that is because, for instance, what if we have the menu $A=(\{1, 2\}, \{2, 3\})$? Then $\{1, 2\}$ should be a selection from it, but does it really contain at most one element from each set in $A$? You could say that it contains two elements from $A_1$ and none from $A_2$. By involving an explicit function $\phi$ that tells you exactly where each element in the selection came from, you can side-step this ambiguity. We make $\phi$ a function over a subset $J$ of the indexing set $I$ so that we can allow for selections that ignore certain of the sets in $A$.
Now, if $A$ is a finite menu and each $A_i$ is finite, and if the members of $A$ are pairwise disjoint, then I believe the number of possible selections from $A$ is $$\prod_{i\in I}(|A_i|+1)$$
My reasoning for this is as follows. Suppose you have to make $n$ choices, and for each choice you have $c_n$ options. How many possible configurations of choices could you make? The answer is obviously $\prod c_i$. When constructing any selection from a menu $A$, you have to make one choice for each $A_i$ - which element you choose from it. You have $|A_i|+1$ options for this choice - $|A_i|$ elements to choose from, with one extra option since you can also choose to not take anything from $A_i$.
