Step 1: Understanding the Concept:
The question requires matching historical and sociological terms with their correct definitions.
Step 2: Detailed Explanation and Matching:
(A) When descent is traced through the mother: This system of kinship is known as (III) Matriliny.
(B) Barbarians or outsiders of the Brahmanical order: In ancient Sanskrit texts, people considered as foreigners or those outside the varna system were referred to as (IV) Mlechchhas.
(C) The first section of Mahabharata: The Mahabharata is composed of 18 sections or books called Parvans. The very first book is the (I) Adi-Parvan (The Book of the Beginning).
(D) Marriage within a kin group, caste, or a group...: The social practice of marrying within one's own specific group is called (II) Endogamy.
Step 3: Final Answer:
The correct pairings are: A-III, B-IV, C-I, and D-II. This combination corresponds to option (4).