(a) Zinc deficiency: Zinc plays a crucial role in maintaining healthy skin. A deficiency in zinc can lead to skin lesions, dryness, and impaired wound healing. Patients on total parenteral nutrition (TPN) are at risk for zinc deficiency if their nutritional support does not adequately replace zinc, making this a highly plausible cause for the described symptoms.

(b) Vitamin A deficiency: Vitamin A is essential for the health of the skin and mucous membranes. Its deficiency can cause dry skin, among other symptoms. Although less directly associated with the specific symptoms described compared to zinc deficiency, it remains a possible cause, especially in the context of inadequate nutritional support.
(c) Vitamin C deficiency: Vitamin C is important for collagen synthesis and can affect skin health. Deficiency might lead to skin symptoms, but it is less commonly associated with dry, scaly, pruritic rashes compared to zinc or vitamin A deficiencies.

Essential fatty acid deficiency: While not listed among the choices, essential fatty acid deficiency can cause skin symptoms, including dryness and scaling. It's a critical diagnosis to consider in patients on TPN because it can have significant health implications if not addressed. The closest option related to this is (d) free fatty acid deficiency, which might be considered in this context, although the terminology is not precise.

Other nutritional deficiencies (e.g., biotin, niacin) can also cause skin rashes but are less likely given the context of TPN and the specific symptoms described. These would be considered rare or less common causes of the patient's presentation.