To solve this problem, we need to understand the properties of an ideal solution.$\\$An ideal solution is defined by the following characteristics:$\\$• The enthalpy of mixing, $\Delta_{mix} H,$ is zero. This means that when two components are mixed,$\\$ there is no heat absorbed or evolved.$\\$• The volume of mixing, $\Delta_{mix} V,$ is also zero. This implies that the total volume of the solution$\\$ is equal to the sum of the volumes of the pure components.$\\$• The entropy of mixing, $\Delta_{mix} S,$ is positive. This is due to the increase in randomness$\\$ when two different substances are mixed.$\\$Let's analyze the given options:$\\$Option 1: $\Delta_{mix} S = 0$ at constant T and P$\\$This is incorrect because the entropy of mixing for an ideal solution is not zero.$\\$It is positive due to the increase in disorder.$\\$Option 2: $\Delta_{mix} V \neq 0$ at constant T and P$\\$This is incorrect because the volume of mixing for an ideal solution is zero.$\\$Option 3: $\Delta_{mix} H = 0$ at constant T and P$\\$This is correct because the enthalpy of mixing for an ideal solution is zero.$\\$Option 4: $\Delta_{mix} G = 0$ at constant T and P$\\$This is incorrect because the Gibbs free energy of mixing is not zero.$\\$It is negative, indicating that the mixing process is spontaneous.$\\$Therefore, the correct option is Option 3: $\Delta_{mix} H = 0$ at constant T and P.