1. **State the problem:** A student must select a two-unit study, choosing one unit from semester one and one unit from semester two. 2. **Identify choices in semester one:** There are 2 mathematics units, 3 language units, and 4 science units. 3. **Calculate total choices in semester one:** $$2 + 3 + 4 = 9$$ 4. **Identify choices in semester two:** There are 2 history units, 3 geography units, and 2 art units. 5. **Calculate total choices in semester two:** $$2 + 3 + 2 = 7$$ 6. **Calculate total number of complete study choices:** Since the student must choose one unit from each semester, multiply the number of choices: $$9 \times 7 = 63$$ 7. **Final answer:** The student has **63** different choices for the complete two-unit study.