Final answer:
The correct sentence, which follows proper punctuation rules, is: 'Basketball leagues were segregated, but unofficial rivalries developed between black and white professional teams.'
Step-by-step explanation:
The correct sentence is: Basketball leagues were segregated, but unofficial rivalries developed between black and white professional teams.
This is corrected by choosing the option that best follows the rules of punctuation and style, which in this case is Option A. The original sentence has a comma before "but," which is a coordinating conjunction linking two independent clauses.
In Option A, the comma before "but" is correctly used, making it the right choice. Options B and C are incorrect; Option B lacks the necessary comma before the conjunction, and Option C incorrectly uses a semicolon followed by 'but.'