1. Problem statement: Given the data: 35 used Longhorn, 20 used Baroque and 17 used Maths Clinic; 9 used Longhorn and Maths Clinic, 3 used Longhorn and Baroque, 8 used Baroque and Maths Clinic only, and 2 used all three. 2. Formula and important rule: Use the inclusion--exclusion principle to find numbers in each Venn region. $$n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(A\cap C)-n(B\cap C)+n(A\cap B\cap C)$$ 3. Interpret the pairwise counts carefully and compute the "only" (exact) two-book overlaps. $$n_{L\&M\,\text{only}}=9-2=7$$ $$n_{L\&B\,\text{only}}=3-2=1$$ $$n_{B\&M\,\text{only}}=8\quad(\text{given as "only"})$$ 4. Compute the numbers who used only one book by subtracting the overlaps and the all-three users from each total. $$\text{Only Longhorn}=35-7-1-2=25$$ $$\text{Only Baroque}=20-1-8-2=9$$ $$\text{Only Maths Clinic}=17-7-8-2=0$$ 5. Check total number of students who used at least one book using inclusion--exclusion and verify consistency. $$35+20+17-9-3-8+2=54$$ $$\text{Total using at least one book}=54$$ 6. Summary (useful grouping for a simple statistics diagram): - Only Longhorn: 25 - Only Baroque: 9 - Only Maths Clinic: 0 - Exactly two (L\&M only): 7 - Exactly two (L\&B only): 1 - Exactly two (B\&M only): 8 - All three: 2 - Total distinct students (at least one book): 54 7. Simple diagram suggestion and display: A bar chart with categories [Only Longhorn, Only Baroque, Only Maths Clinic, L\&M only, L\&B only, B\&M only, All three] and heights [25,9,0,7,1,8,2] visualises the grouping clearly and helps decision making. 8. Which book to replace and why: Replace the Maths Clinic book. Reason: the computed count of students who used Maths Clinic only is zero, i.e. no student read the Maths Clinic book alone, so it contributes least to exclusive adoption and is the weakest standalone resource. $$\text{Only Maths Clinic}=0\;\Rightarrow\;\text{replace Maths Clinic}\,.$$ 9. Probability questions and missing data note: There is not enough information in the given statement to compute the probability that a student "failed" because the dataset gives book-usage counts, not pass/fail marks. If by "failed" the teacher means "used none of the three books" and the total class size is $N$, then the probability that a randomly selected student used none is $$P(\text{none})=\dfrac{N-54}{N}\,.$$ If the class size were 60 as an example, then the number using none is $60-54=6$ and $$P(\text{none})=\dfrac{6}{60}=\dfrac{\cancel{6}}{\cancel{60}}=\dfrac{1}{10}\,.$$ 10. Final practical defence for the head teacher: The Venn grouping shows Maths Clinic has zero exclusive users, so replacing it with Fountain publisher is justified because it had no sole adopters and thus low independent demand.