Assigning Balls to Bins with Constraints on Which Ball to Go to Which Bin? Let us say we have $m$ bins and $n$ balls. Every bin $i$ has capacity $c_i$ which is the number of balls that can be put into bin $i$. We have $c_i\geq1$ for all $i$. For each bin $i$, there is a collection of sets $S_i=\{X_1,X_2,\ldots,X_{k_i}\}$ for given $k_i$. Each $X_j\in S_i$ is the set of balls that can be put into bin $i$. We have $|X_j|\leq c_i$ and $\emptyset\in S_i$ for all $i$.

For example, for $m=2$ and $n=3$, with $c_1=1$ and $c_2=2$, say we have $k_1=4$ and $k_2=5$. Say we have $S_1=\{\emptyset,\{1\},\{2\},\{3\}\}$. $S_2=\{\emptyset,\{1\},\{2\},\{3\},\{2,3\}\}$. This means that ball $1$, $2$ or $3$ can be each assigned to bin $1$. Also, each ball can be assigned to bin $2$. Further, balls $2$ and $3$ can be together assigned to bin $2$. We might have an instance with $k_2=6$ and $S_2=\{\emptyset,\{1\},\{2\},\{3\},\{2,3\},\{1,2\}\}$ for example. 

We want to assign the maximum number of balls into the bins. Is this easy or hard?
You can reduce the Exact Cover problem to your problem. The elements in the Exact Cover problem corresponds to the balls in your problem. For each subset $T$ in the Exact Cover problem, we construct a bin $i$ with $S_i=\{\emptyset, T\}$. Then there exists an exact cover if and only if all balls can be put into the bins. Hence, your problem is NP-hard.