Let $p$ be a positive integer. For each positive integer $N$, let: $$F: \{1, \ldots , pN\} \rightarrow \{1, \ldots, N\} $$ $$ F(n) = \lceil n/p \rceil$$ Let $r \in [0,1]$. I'm curious about the limit of the expected value $\mathbb E(\#F(A)/N)$ for $A \subseteq \{1, \ldots, pN\}$ a subset of cardinality $\lfloor rpN\rfloor$ selected randomly. I have no background in combinatorics; this question may be below the level of mathoverflow.