  Let $\lambda$ be a partition of the positive integer $n$, selected uniformly
at random among all such partitions. Corteel et al. (1999) proposed three
different procedures of sampling parts of $\lambda$ at random. They obtained
limiting distributions of the multiplicity $\mu_n=\mu_n(\lambda)$ of the
randomly-chosen part as $n\to\infty$. The asymptotic behavior of the part size
$\sigma_n=\sigma_n(\lambda)$, under these sampling conditions, was found by
Fristedt (1993) and Mutafchiev (2014). All these results motivated us to study
the relationship between the size and the multiplicity of a randomly-selected
part of a random partition. We describe it obtaining the joint limiting
distributions of $(\mu_n,\sigma_n)$, as $n\to\infty$, for all these three
sampling procedures. It turns out that different sampling plans lead to
different limiting distributions for $(\mu_n,\sigma_n)$. Our results generalize
those obtained earlier and confirm the known expressions for the marginal
limiting distributions of $\mu_n$ and $\sigma_n$.
