Both are correct. The first sentence using "with whom" is a very formal style.
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RegardsThis one would be better in academic writing:If I had to choose one girl with whom to go out, it would be she. Yes, I know using she instead of her may sound strange to you and maybe incorrectIt is not incorrect. It is just, as you said, very stilted.
RegardsI much prefer the second sentence, which is both grammatically sound an