Permutation of Groups - looking for the right term

Question

I'm looking for more detailed information about the following problem, but i'm missing a right keyword, or term for this:

Let's assume i have 10 people and they are assigned to groups:

person    group
0         A
1         A
2         A
3         B
4         B
5         B
6         C
7         C
8         D
9         D

i want to assign each person to another one, but that person must not be within the same group. How many different groupings (seen as a whole) are possible here?

I thought it would be the right way to use combinations without repetition: $$ \frac{10!}{3!\cdot 3!\cdot 2!\cdot 2!} $$ is that the right way?

What i finally want do achieve is, to create an algorithm, that checks whether a fully assignment (each person is connected to another one) is possible.

e.g.

1    A
2    A
3    A
4    B
5    C

is not possible, as the 3. A can not be assigned to another group.

but (with sample-assignment):

1    A      |    1 -> 3    
2    A      |    2 -> 4
3    B      |    3 -> 2
4    B      |    4 -> 5
5    C      |    5 -> 1

is possible.

Answer 1

Construct a graph whose vertices are the people, and where an edge connects two people if they are in different groups. The assignment is possible if and only if this graph has a perfect matching (that is, a subset of edges such that each vertex is a member of exactly one chosen edge).

Counting the number of perfect matchings (your first question) as well as finding a maximal matching (which may be perfect if there is one) are both discussed in the Wikipedia article here. Neither is easy, though finding a matching appears to have complexity of the order of $VE$ where $V$ and $E$ are the number of vertices and edges in the graph.