Step 1: The triad in a male child is eczema (the rash shown), thrombocytopenia (low platelets) and recurrent infections. This classic triad in a boy points to Wiskott-Aldrich syndrome.
Step 2: It is an X-linked disorder from a mutation in the WAS gene (WASP protein) at Xp11.23, which is why it affects males. The platelets are characteristically small in size. There is progressive loss of T lymphocytes with poor antibody response to polysaccharide antigens, so cellular and humoral immunity both fail.
Step 3: Job's syndrome (hyper-IgE) gives recurrent cold abscesses and very high IgE but normal platelets. Chediak-Higashi causes partial albinism and giant granules in leukocytes, not eczema with thrombocytopenia. The full eczema + thrombocytopenia + infection triad fits only Wiskott-Aldrich.
Ref: Robbins and Cotran Pathologic Basis of Disease, 9th Edition, Page 242.