To determine which ion exhibits both $d-d$ transition and paramagnetism, we need to consider the electronic configuration and the oxidation state of the central metal ion in each given option.
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$MnO_4^{2-}$: The manganese ion in $MnO_4^{2-}$ is in the +6 oxidation state. In this state, manganese has the electronic configuration [Ar] 3d1 4s0. The presence of one unpaired electron in the 3d orbital means the ion is paramagnetic. The $d-d$ transition is possible since the unpaired electron in the d orbital can undergo electronic transition.
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$CrO_4^{2-}$: The chromium ion in $CrO_4^{2-}$ is in the +6 oxidation state, corresponding to a 3d0 configuration. Since there are no unpaired electrons, it is not paramagnetic, and $d-d$ transitions are not possible.
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$MnO_4^{-}$: The manganese in $MnO_4^{-}$ is in the +7 oxidation state. Manganese in this state also has a 3d0 configuration, leading to no unpaired electrons and hence no paramagnetism or $d-d$ transitions.
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$Cr_2O_7^{2-}$: The chromium in $Cr_2O_7^{2-}$ is in the +6 oxidation state, similar to $CrO_4^{2-}$ with a 3d0 configuration. It does not exhibit paramagnetism or $d-d$ transitions either.
After analyzing each option, $MnO_4^{2-}$ is the only ion that satisfies the conditions of having a $d-d$ transition due to the presence of an unpaired electron, as well as displaying paramagnetism.
Thus, the correct answer is $MnO_4^{2-}$.