  The Stirling number of the second kind $n\brace k$ counts the number of ways
to partition a set of $n$ labeled balls into $k$ non-empty unlabeled cells. As
an extension of this, we consider $b_1+b_2+\ldots+b_n$ balls with $b_1$ balls
labeled $1$, $b_2$ balls labeled $2$, $\ldots$, $b_n$ balls labeled $n$ and
$c_1+c_2+\ldots+c_k$ cells with $c_1$ cells labeled $1$, $c_2$ cells labeled
$2$, $\ldots$, $c_k$ cells labeled $k$ and then we called the number of ways to
partition the set of these balls into non-empty cells of these types as the
\textit{mixed partition numbers}. As an application, we give a new statement of
the $r$-Stirling numbers of second kind and $r$-Bell numbers. We also introduce
the \textit{$r$-mixed Stirling number of second kind and $r$-mixed Bell
numbers}. Finally, for a positive integer $m$ we evaluate the number of ways to
write $m$ as the form $m_1\cdot m_2\cdot\ldots\cdot m_k$, where $k\geqslant 1$
and $m_i$'s are positive integers greater than $1$.
