Step 1: Conceptual Grasp:
The task involves associating historical and sociological terminology with their accurate definitions.
Step 2: Detailed Analysis and Association:
(A) Tracing lineage through the female parent: This kinship structure is identified as (III) Matriliny.
(B) Individuals outside the Brahmanical social hierarchy: Ancient Indian texts designate those considered foreigners or outside the established varna system as (IV) Mlechchhas.
(C) The inaugural part of the Mahabharata: The epic Mahabharata comprises 18 books known as Parvans. The primary book is the (I) Adi-Parvan, signifying 'The Book of the Beginning'.
(D) Marriage exclusively within a designated social unit: The practice of marrying within one's own defined kin group, caste, or community is termed (II) Endogamy.
Step 3: Conclusive Pairing:
The accurate matches are A-III, B-IV, C-I, and D-II. This configuration aligns with option (4).