1. **State the problem:** We need to verify if the given data about students' learning mode preferences is consistent. 2. **Given data:** - Total students surveyed: $111$ - Face-to-Face (F): $27$ - Online (O): $19$ - Modular (M): $15$ - Face-to-Face and Online (F \cap O): $17$ - Face-to-Face and Modular (F \cap M): $13$ - Online and Modular (O \cap M): $11$ - All three modes (F \cap O \cap M): $9$ 3. **Formula and rules:** The principle of inclusion-exclusion for three sets states: $$|F \cup O \cup M| = |F| + |O| + |M| - |F \cap O| - |F \cap M| - |O \cap M| + |F \cap O \cap M|$$ 4. **Calculate the union:** $$|F \cup O \cup M| = 27 + 19 + 15 - 17 - 13 - 11 + 9$$ 5. **Simplify step-by-step:** $$= (27 + 19 + 15) - (17 + 13 + 11) + 9$$ $$= 61 - 41 + 9$$ $$= 20 + 9$$ $$= 29$$ 6. **Interpretation:** The total number of students who prefer at least one mode is $29$, but the total surveyed is $111$. This suggests a discrepancy because the union should be equal to or less than the total surveyed, but here it is much less. 7. **Check for consistency:** Since $|F \cup O \cup M| = 29$ but total students surveyed is $111$, the data is inconsistent or incomplete. **Final answer:** The data is not correct or consistent based on the principle of inclusion-exclusion.