1. **Problem (a):** In a test, 20 pupils passed in Science or Mathematics. If 5 passed both subjects and 3 more passed Mathematics than Science, find the number of pupils who passed in each subject. 2. **Formula and rules:** Use the principle of inclusion-exclusion for two sets: $$|S \cup M| = |S| + |M| - |S \cap M|$$ where $|S|$ is the number of pupils passing Science, $|M|$ is the number passing Mathematics, and $|S \cap M|$ is the number passing both. 3. **Step-by-step solution:** - Let $|S| = x$ (number passing Science). - Then $|M| = x + 3$ (3 more passed Mathematics). - Given $|S \cup M| = 20$ and $|S \cap M| = 5$. 4. Substitute into the formula: $$20 = x + (x + 3) - 5$$ 5. Simplify: $$20 = 2x + 3 - 5$$ $$20 = 2x - 2$$ 6. Solve for $x$: $$2x = 22$$ $$x = 11$$ 7. Find $|M|$: $$|M| = 11 + 3 = 14$$ 8. **Answer:** - Number passing Science: $11$ - Number passing Mathematics: $14$