The correct answer is option 1: I, II and III.

Let`s dissect each statement to understand why it is correct:

I. The first statement says that rainy tropical climate regions usually extend not more than five to ten degrees on either side of the equator. This is true because these regions are typically found in the vicinity of the equator, where the warm ocean temperatures provide the necessary moisture for rainfall.

II. The second statement states that rainy tropical climate regions are characterized by constant high temperature and humidity with little range. This is also true because these regions experience consistently high temperatures throughout the year, typically ranging from 25 to 35 degrees Celsius, and high humidity levels due to the abundant moisture in the air.

III. The third statement mentions that these regions have the sun overhead or nearly so throughout the year. This is true because of their proximity to the equator, where the sun`s rays are more direct, resulting in a higher sun angle and more intense sunlight.

Therefore, all three statements accurately describe the characteristics of rainy tropical climate regions, making option 1 the correct answer.