1. ## a combinatorics problem

I have B different labels and C different objects. Each label can be assigned to more than one object at the same time, and each object can have more than one label.

I want to assign every label to exactly X of my C objects, while ensuring that no object has more than Y different labels.

For some values of C, this will not be possible. My question is: What is the smallest value of C for which I can do this assignment?

2. ## my solution

...is in the next post....

3. ## my solution

I think the lowest number of objects is given by

$X$ if $B \leq Y$
and $X+(B-Y)\left\lceil X/Y \right\rceil$ otherwise

where with $B \leq Y$ we have every object getting all labels, and so X objects are required, while otherwise $\left\lceil X/Y \right\rceil$ extra objects are required to be assigned the labels not used up by the first X objects.