Which of the compounds can exist together?

The correct option is "FeCl₂, SnCl₂".

To determine which pair of compounds can exist together in a solution, we must analyze the possibility of a chemical reaction between them. Specifically, we look for potential oxidation-reduction (redox) reactions where one species can oxidize the other.
If two substances can undergo a redox reaction, they cannot coexist in a stable solution. If no redox reaction is thermodynamically favorable between them, they can exist together.

Let's analyze the stability and oxidation states of the metals in the options:

1. HgCl₂ and SnCl₂:
Mercury in ${\mathrm{HgCl}}_2$ is in the +2 oxidation state (${\mathrm{Hg}}^{2+}$), which is a strong oxidizing agent. Tin in ${\mathrm{SnCl}}_2$ is in the +2 oxidation state (${\mathrm{Sn}}^{2+}$), which is a well-known reducing agent because it readily oxidizes to the more stable +4 state (${\mathrm{Sn}}^{4+}$). Therefore, they react as follows:
$2{\mathrm{HgCl}}_2+{\mathrm{SnCl}}_2\to {\mathrm{Hg}}_2{\mathrm{Cl}}_2+{\mathrm{SnCl}}_4$
Because they react with each other, they cannot exist together.

2. FeCl₃ and KI:
Iron in ${\mathrm{FeCl}}_3$ is in the +3 oxidation state (${\mathrm{Fe}}^{3+}$), which is moderately oxidizing. Iodide ($I^{-}$) in KI is a reducing agent. ${\mathrm{Fe}}^{3+}$ easily oxidizes $I^{-}$ to elemental iodine ($I_2$) while being reduced to ${\mathrm{Fe}}^{2+}$:
$2{\mathrm{FeCl}}_3+2\mathrm{KI}\to 2{\mathrm{FeCl}}_2+2\mathrm{KCl}+I_2$
Thus, they react and cannot exist together.

3. FeCl₃ and SnCl₂:
Again, ${\mathrm{Fe}}^{3+}$ is an oxidizing agent and ${\mathrm{Sn}}^{2+}$ is a reducing agent. ${\mathrm{Fe}}^{3+}$ oxidizes ${\mathrm{Sn}}^{2+}$ to ${\mathrm{Sn}}^{4+}$ and is itself reduced to ${\mathrm{Fe}}^{2+}$:
$2{\mathrm{FeCl}}_3+{\mathrm{SnCl}}_2\to 2{\mathrm{FeCl}}_2+{\mathrm{SnCl}}_4$
Thus, they react and cannot exist together.

4. FeCl₂ and SnCl₂:
Here, iron is in the +2 oxidation state (${\mathrm{Fe}}^{2+}$) and tin is in the +2 oxidation state (${\mathrm{Sn}}^{2+}$). Both ${\mathrm{Fe}}^{2+}$ and ${\mathrm{Sn}}^{2+}$ act primarily as reducing agents (they prefer to be oxidized to ${\mathrm{Fe}}^{3+}$ and ${\mathrm{Sn}}^{4+}$ respectively). Neither of them has a strong enough oxidizing potential under standard conditions to oxidize the other. Consequently, no redox reaction occurs between them, meaning they do not react and can stablely exist together in solution.