Is it correct to say “a lot of people in my hometown have never even drunk coffee”? Somehow the word drunk sticks out and I’m not sure if I should use “drank” or “drunk”.

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Your sentence is absolutely correct as it is. Great job, shia!
BTW, is that true about your hometown? Where is that place?

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I'd say many people instead...

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“Have done something”
You need to use past participle “drunk”here.

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Yes, the Past Participle of "drink" is "drunk."