Could I get a residency at a hospital where I didn't do rotations?

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Yes. Apply.
Depending on the specialty and the program, some give preference to people who do a rotation there. Some don't. So, don't worry and apply. You can show more interest and get to know faculty, but you could also hurt your chances if you don't make a good impression or don't click. IN either of the last two cases, you may decide that the program is not best suited for you.
 
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