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The word 'but' follows the same rules as any other conjunction, just like 'and' or 'or'. It's a word that joins two phrases. All of your examples are correct. The first phrase ends with a comma, ...
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#3: Attribution notice added
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#2: Initial revision
The word 'but' follows the same rules as any other conjunction, just like 'and' or 'or'. It's a word that joins two phrases. All of your examples are correct. The first phrase ends with a comma, and the second phrase starts with the conjunction. Your first two sentences follow this rule perfectly. There is no comma in the third sentence, and as far as I know, neither should there be. Based on your wording, you might be thinking that the word 'but' requires a comma at all times. It does not. Treat it like any other conjunction.