1. **Problem Statement:** In a bootcamp of 60 students: - 35 students can code in Python. - 40 students can code in Java. - 10 students can code in neither Python nor Java. Find: 1. Number of students who code in both languages. 2. Number of students who code in exactly one language. 2. **Formula and Rules:** We use the principle of inclusion-exclusion for two sets $A$ and $B$: $$|A \cup B| = |A| + |B| - |A \cap B|$$ where: - $|A|$ is the number of students coding in Python. - $|B|$ is the number of students coding in Java. - $|A \cap B|$ is the number coding in both. - $|A \cup B|$ is the number coding in at least one language. 3. **Step-by-step Solution:** - Total students = 60 - Students coding neither = 10 - Therefore, students coding at least one language: $$|A \cup B| = 60 - 10 = 50$$ - Using inclusion-exclusion: $$|A \cup B| = |A| + |B| - |A \cap B|$$ Substitute known values: $$50 = 35 + 40 - |A \cap B|$$ - Solve for $|A \cap B|$: $$|A \cap B| = 35 + 40 - 50 = 25$$ - Number coding in exactly one language is: $$|A| + |B| - 2|A \cap B| = 35 + 40 - 2 \times 25 = 75 - 50 = 25$$ 4. **Final answers:** - Students coding in both languages: **25** - Students coding in exactly one language: **25**