Number of ways to select numbers, each 1 from different lists without repetition I want the numbers of ways to select numbers each 1 from different lists without allowing repetition.
Eg-


*

*List 1 : 5, 100, 1 

*List 2 : 2

*List 3 : 5, 100

*List 4 : 2, 5, 100


I want to select 1 number from each, without repetition.
Answer in this case will be : 2 ( 1,2, 5,100 and 1,2,100,5). 

After thinking a lot i am getting the feeling of a direct formulae of Permutation and Combination, But i cant figure it out. 
Any genius out there?
 A: Let $V_1,V_2\dots V_n$ be the sets and $U$ be the union of the sets. for each $u\in U$ construct the set $S_u$ consisting of the elements of the $V$'s that contain $u$. The answer is therefore the same as the number of partitions of the set $\{V_1,V_2\dots V_n\}$ in which all its parts are of the form $C_u$ for some $u\in U$.
The simple problem of finding out if there is at least one solution to this is called the exact cover problem, this problem is known to be $NP$ complete. So the problem you are asking for must be at least as "hard" as this problem
A: This counting problem is $\#P$-complete.
We construct a $(0,1)$-matrix with (a) rows indexed by list number, (b) columns indexed by the possible elements, and (c) a $1$ in row $\ell$ and column $x$ if and only if $x$ occurs in list $\ell$.
Finding the number of combinations is the same as computing the permanent of this $(0,1)$-matrix (actually, it's a generalization of it, to rectangular $(0,1)$-matrices), which is the canonical $\#P$-complete problem.
Informally, an answer other than "it's hard" would imply $P=NP$.
