I've been struggling with this problem for a while now, but I'm not sure how to proceed :
Given a set of $n$ distinct objects, we seek to find the number of ways we can distribute all of the elements belonging to this set to $3$ kinds of people.
$a$ of them want to get an odd number of objects, $b$ of them want to get an even number of objects and $c$ of them don't care about the parity of the objects they receive.
In how many ways can this division be performed, such that everybody is satisfied? Some of the people may not receive an object.
Since this problem has no label in my textbook, I'm not sure how to place it in a category, but I'm fairly certain it's a combinatorial problem.
I've been thinking by writing the solution as a sum of products of binomial coefficients, but I haven't been able to get an useful form. I guess that each term would be comprised by two factors, where the first one would mean in how many ways we can get an odd number of objects and the second one would be the number of ways we can choose an even number of objects from the remaining ones.
How would you solve this?