1. **State the problem:** We have 50 children in total. 28 attend piano lessons, 17 attend flute lessons, and 12 attend neither. We need to find how many attend only piano lessons. 2. **Use the formula for sets:** Let $P$ be the set of children attending piano lessons, $F$ be the set attending flute lessons, and $N$ be those attending neither. Total children: $$|P \cup F| + |N| = 50$$ Given: $$|P| = 28, \quad |F| = 17, \quad |N| = 12$$ 3. **Calculate the number attending piano or flute:** $$|P \cup F| = 50 - 12 = 38$$ 4. **Use the inclusion-exclusion principle:** $$|P \cup F| = |P| + |F| - |P \cap F|$$ Substitute known values: $$38 = 28 + 17 - |P \cap F|$$ 5. **Solve for the intersection:** $$|P \cap F| = 28 + 17 - 38 = 7$$ 6. **Find the number attending only piano lessons:** $$|P \text{ only}| = |P| - |P \cap F| = 28 - 7 = 21$$ **Final answer:** 21 children attend only piano lessons.