**Core Concept**
The development of external genitalia in a fetus is a complex process influenced by the presence or absence of androgens, specifically dihydrotestosterone (DHT). The SRY gene on the Y chromosome triggers testis development, which then produces Müllerian-inhibiting substance and androgens. The presence of androgens leads to the development of male external genitalia.

**Why the Correct Answer is Right**
The correct answer is linked to the formation of the genital tubercle, which eventually develops into the penis in males and the clitoris in females. The presence of androgens, specifically DHT, is crucial in the differentiation of the genital tubercle into a penis. DHT acts on the androgen receptor, which is present in the genital tubercle, to induce the development of male external genitalia. This process occurs around 8-10 weeks of gestation.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect because the formation of external genitalia is not primarily influenced by the presence of estrogens, which are more involved in the development of female secondary sexual characteristics.
**Option B:** This option is incorrect because the development of external genitalia is not solely dependent on the presence of Müllerian-inhibiting substance, which is produced by the testes and prevents the development of the Müllerian ducts in males.
**Option C:** This option is incorrect because the formation of external genitalia is not primarily influenced by the presence of cortisol, which is a glucocorticoid involved in various physiological processes.

**Clinical Pearl / High-Yield Fact**
It's essential to remember that the presence of androgens, particularly DHT, is critical in the development of male external genitalia. This knowledge is crucial in understanding the pathophysiology of disorders such as androgen insensitivity syndrome.

**Correct Answer: C. 8-10 weeks of gestation**