Match Column-$I$ with Column-$II$.

Column-$I$	Column-$II$
$(1)$ $\frac{{{m_1}{m_2}}}{{{m_1} + {m_2}}}$	$(a)$ Reduced mass of a two-particle system
$(2)$ $\frac{{{r_1} + {r_2}}}{2}$	$(b)$ Position vector of the center of mass for a system of two equal masses

(A) For $(1)$, the reduced mass $\mu$ of a system of two particles with masses $m_1$ and $m_2$ is defined as $\mu = \frac{m_1 m_2}{m_1 + m_2}$. Thus, $(1)$ matches with $(a)$.
For $(2)$, the center of mass $R_{cm}$ of a system of two particles with masses $m_1$ and $m_2$ at positions $r_1$ and $r_2$ is given by $R_{cm} = \frac{m_1 r_1 + m_2 r_2}{m_1 + m_2}$. If $m_1 = m_2 = m$, then $R_{cm} = \frac{m(r_1 + r_2)}{2m} = \frac{r_1 + r_2}{2}$. Thus, $(2)$ matches with $(b)$.
The correct matching is $(1-a, 2-b)$.