To determine the number of unpaired electrons in the $Mn^{2+}$ ion, let's first consider the electron configuration of a neutral manganese (Mn) atom.
Manganese (Mn) has an atomic number of 25, so the electron configuration of neutral Mn is:
$[Ar] 4s^2 3d^5$
When manganese forms a $Mn^{2+}$ ion, it loses two electrons. These electrons are removed first from the 4s orbital because it is higher in energy than the 3d orbital.
So, the electron configuration of $Mn^{2+}$ is:
$[Ar] 3d^5$
The 3d subshell has 5 electrons in the case of $Mn^{2+}$, and since the 3d orbitals can hold a maximum of 10 electrons, the 5 electrons will occupy the 5 available d orbitals with each orbital having 1 electron (this is the maximum number of unpaired electrons).
Therefore, the number of unpaired electrons in $Mn^{2+}$ is:
Answer: 5