A group of two people, or in golf, a match between two players.
From 'two' plus the suffix '-some' (forming nouns meaning a group of that many, as in 'quartet' or 'group'). This is the same suffix used in 'awesome' and 'handsome'.
The '-some' suffix is really versatile—'twosome', 'threesome', 'foursome', 'awesome', 'lonesome', 'handsome'—they all use the same ancient ending that originally meant 'characterized by' or 'full of', which shows how English recycles pieces to build new words.
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