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The school book and Wikipedia state that coordination isomers are:

In a coordination isomer the total ratio of ligand to metal remains the same, but the ligands attached to a specific metal ion change.

For example, a solution containing $\ce{[Co(NH3)6]^3+}$ and $\ce{[Cr(CN)6]^3−}$ is a coordination isomer with a solution containing $\ce{[Cr(NH3)6]^3+}$ and $\ce{[Co(CN)6]^3−}$.

So in this definition, consider $\ce{[Co(NH3)5(CN)][Cr(CN)5(NH3)]}$. Is it coordination isomer of $\ce{[Co(NH3)6][Cr(CN)6]}$ or not?

My teacher says yes, but I'm not quite sure according to Wikipedia.

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Yes, it is. In both the cases, the number of ligands remains same, which is according to the definition in Wikipedia.

For any type of structural isomerism, the premise is that the primary valence and secondary valence (as defined by Werner) of central metal/ ion remains same for all the isomers.

In this case, the primary valence is 3 and secondary valence is 6, which remains same for all isomers.

Other possible coordination isomers are:

$\ce{[Co(NH3)4(CN)2]+[Cr(NH3)2(CN)4]-}$

$\ce{[Cr(NH3)4(CN)2]+[Co(NH3)2(CN)4]-}$

$\ce{[Cr(NH3)5(CN)]^2+[Co(NH3)(CN)5]^2-}$

(cation is always written first followed by anion)

I don't think $\ce{[Co(NH3)3(CN)3][Cr(NH3)3(CN)3]}$ can exist because, now both parts become neutral.

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