Combination of fixed size from a list of list I would like to find how many combinations of a fixed size $k$ exist into a list of list. 
Where the list containing others is of size $n$.
Where size of inner lists are $s_1, s_2,..., s_n$
Each combination can only have one element of inner lists.
We suppose $k \le s_{min}$ where $s_{min}$ is the lowest cardinality of inner lists.
Example with $k=2$, $n=3$, $s_1=3$, $s_2=2$, $s_3=2$
list[list[a,b,c], list[1,2], list[$,€]]

Gives
(a,1), (a,2), (b,1), (b,2), (c,1), (c,2), (a,€), (a,$), (b,€), (b,$), (c,€), (c,$), (1,€), (1,$), (2,€), (2,$)

 A: Lets assume that there are no identical members in any inner list, and no identical elements that are members of two different lists. Outer list has $n$ members (inner lists), $s_i$ is number of members of $i$-th inner list. Pick any (strictly) increasing sequence $1\le i_1<i_2<..<i_k\le n$ and assume these are indexes of  inner lists from which members will be drawn. Member of each lists is picked independently, so for fixed indexes $1\le i_1<i_2<..<i_k\le n$ the number of possible draws is $s_{i_1}*..*s_{i_k}$. This gives the sum:
$$ \sum_{1\le i_1<i_2<..<i_k\le n}s_{i_1}*..*s_{i_k}$$
over all possible sequences to be the answer. In your example $3*2+3*2+2*2$.
Special case: if $s_1=..=s_n$, then:
$$\sum_{1\le i_1<i_2<..<i_k\le n}s_{i_1}*..*s_{i_k}=\sum_{1\le i_1<i_2<..<i_k\le n} s_{1}^{\ k}=\binom{n}{k}s_1^{\ k}$$
because number of strictly increasing sequences of length $k$ with elements from {1,..,n} is $\binom{n}{k}$.
A: In your example all possible lists of all possible lengths (including $0$) may be viewed as a generating function
$$(1+\mathbf{a}+\mathbf{b}+\mathbf{c})(1+\mathbf{1}+\mathbf{2})(1+\mathbf{\$}+\mathbf{€})$$
Since all ordered lists will appear in the expanded polynomial. Note that list items are shown in bold.
As you only care about the length of the list (specifically in your example you have use length $2$) then we can call all symbols $x$ and let the index of $x$ enumerate the length of the list and the coefficient of $x^k$ enumerate the number of lists length $k$. 
In your example we have
$$(1+x+x+x)(1+x+x)(1+x+x)=(1+3x)(1+2x)^2$$
and to count lists of length $2$ we want the coefficient of $x^2$ in this expansion
$$(1+3x)(1+2x)^2=12 \, x^{3} + 16 \, x^{2} + 7 \, x + 1$$
which is $16$.
In general for a set of $n$ lists with $s_1, s_2, \ldots, s_n$ items we can enumerate the number of lists length k by evaluating the $x^k$ coefficient of
$$\prod_{i=1}^{n}(1+s_ix)$$ 
we represent this with the "find coefficient of $x^k$" operator $\left[x^k\right]$
$$\text{lists of length $k$}=\left[x^k\right]\prod_{i=1}^{n}(1+s_ix)$$
Note that, although $x$ "looks like" a variable, it is in fact an indeterminate meaning that we use it purely for it's multiplicative and additive properties in order to determine coefficients, we never even consider the question of evaluating $x$. 
