I have a set $S$ of $n$ elements. I select $r$ elements from the set $E_1, E_2, E_3, \ldots, E_r$ that may not be all distinct. I select again $r$ elements from the set $F_1,F_2,F_3,\ldots, F_r$ in a similar manner.
What is the probability $P$ that $E_i = F_j$ for some $i$ and $j$?
I have to prove that this probability is at least one half of the probability that $2r$ randomly chosen elements selected from $S$ are not all distinct.