TITLE: How to pick all the colors?
QUESTION [6 upvotes]: Prove of disprove: Suppose there are n boxes, each containing m balls of the same color,with n colors in total. No matter how we reallocate these balls (still each box contains m balls), we can pick up a ball from each box such that all colors are picked.
This question came to my mind when I tried to solve a problem in algebra: Let $H $ be a subgroup of the finite group G. Show that there exists a subset {$z_1,...,z_r$} of G which is simultaneously a set of representatives of the left and of the right cosets of H in G, that is, G=$\cup_{i=1}^r z_iH=\cup_{i=1}^rHz_i$.

REPLY [2 votes]: As Gerry Myerson pointed out in the comment, it is an application of Hall's theorem. The statement in terms of systems of distinct representatives is the following:
Given a universe $U$ of $n$ elements and a collection of $n$ sets, we want to pick one element from each set so that all elements are covered. Then a sufficient condition is that the union of any $k$ sets contains at least $k$ elements.
In the re-assignment, let each box be identified with the set of colors it contains, so that $U$ is the set of colors. It is easy to see that the union of any $k$ boxes must contain at least $k$ colors; otherwise, the number of elements would be less than $km$.