Transition metals can exhibit variable valency. It is because of (a) the smaller atomic radius (b) the high screening effect (c) the very less energy difference between \((\mathrm{n}-1)\) d-subshell and ns-subshell (d) the high nuclear charge

Transition metals are elements found in groups 3 to 12 of the periodic table. These metals have partly filled d-subshells, which is the reason they exhibit variable valency. Valency refers to the combining capacity of an element to form compounds.

Let us analyze the given options: (a) the smaller atomic radius: This option is not directly related to the variable valency of transition metals. Transition metals have smaller atomic radius due to the increasing nuclear charge but not exclusively responsible for variable valency. (b) the high screening effect: The screening effect refers to the phenomenon where the outer electrons feel less force from the nucleus due to the already existing inner electrons, which repel the outer ones. While it might have some effect on the electronic properties of the transition metals, it's not the main reason for their variable valency. (c) the very less energy difference between \((\mathrm{n}-1)\) d-subshell and ns-subshell: This option suggests that the less energy difference between the orbitals can be the reason for the variable valency. Indeed, the small energy gap allows the electrons to jump between the \((\mathrm{n}-1)\) d-subshell and the ns-subshell, resulting in variable oxidation states. (d) the high nuclear charge: While transition metals indeed have higher nuclear charges compared to other elements, this is not the main reason for their variable valency. The nuclear charge may impact other aspects of the transition metals, but it is not directly responsible for their variable valency.

After analyzing each option, we can conclude that the correct answer is (c) the very less energy difference between \((\mathrm{n}-1)\) d-subshell and ns-subshell. This energy difference allows the electrons to move between the orbitals, giving rise to different oxidation states and ultimately leading to the variable valency observed in transition metals.