I need some hints on this... So I have multisets (sets where the same value can occur more than once) consisting of nonnegative integers $x$, with $0 \le x < p$. I would like to prove that if I multiply, modulo $p$, every nonzero value in one set by every nonzero value in another set that their frequencies or occurences tend to level out.
In other words, say modulo $3$, we have three multisets $\{1,2,2,2\}$, $\{1,2,2,2\}$, $\{1,2,2,2\}$. If we take every possible multiplication of elements of the three sets, we end up with $28$ ones and $36$ twos. These sets are more balanced... There is, in some sense, nearly an equal amount of ones and twos.
I want to prove this myself, but I need some hints or clues. I thought that maybe I could even cite some idea or theory, such as the central limit theorem. Even if you just tell me the topic to study, I would appreciate it.