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How many integers between $1$ and $60$ contain the digit $3$ at least once?

(A) $6$ (B) $10$ (C) $14$ (D) $15$ (E) $20$

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project. Solutions are the publisher's, linked not copied.