**Core Concept**
The child's symptoms suggest an underlying condition characterized by autoimmune manifestations, endocrine dysfunction, and mucocutaneous involvement.

**Why the Correct Answer is Right**
The child's presentation is classic for DiGeorge Syndrome, also known as 22q11.2 deletion syndrome, which is caused by a microdeletion in the long arm of chromosome 22. This condition leads to thymic hypoplasia, resulting in impaired T-cell development and function. The resultant immunodeficiency increases the risk of autoimmune diseases, such as alopecia and psoriasis. Additionally, the deletion affects the parathyroid glands, leading to hypoparathyroidism, which can cause hypocalcemia and hyperpigmentation. The syndrome also affects the hypothalamic-pituitary-adrenal axis, resulting in hypogonadism.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect because the child's symptoms do not suggest a primary autoimmune disorder, such as lupus or rheumatoid arthritis.

**Option B:** This option is incorrect because the child's symptoms are not consistent with a primary endocrine disorder, such as thyroiditis or adrenal insufficiency.

**Option C:** This option is incorrect because the child's symptoms are not consistent with a primary skin condition, such as eczema or acne.

**Clinical Pearl / High-Yield Fact**
DiGeorge Syndrome is a classic example of a condition that can present with a triad of symptoms: congenital heart disease, cleft palate, and hypocalcemia.

**Correct Answer:** C. DiGeorge Syndrome