1. **Problem Statement:** A group of students used three different books: Longhorn, Baroque, and Maths Clinic. Given: - 9 used Longhorn and Maths Clinic - 3 used Longhorn and Baroque - 8 used Baroque and Maths Clinic only - 2 used all three books - No student used Fountain publisher alone We need to group the marks, identify which book to replace, and find the probability a student failed. 2. **Step 1: Define sets and use inclusion-exclusion principle** Let: $L$ = students using Longhorn $B$ = students using Baroque $M$ = students using Maths Clinic Given: $|L \cap M| = 9$ $|L \cap B| = 3$ $|B \cap M| = 8$ $|L \cap B \cap M| = 2$ 3. **Step 2: Calculate number of students using exactly two books** Students using exactly two books = $|L \cap M| + |L \cap B| + |B \cap M| - 3 \times |L \cap B \cap M|$ $$= 9 + 3 + 8 - 3 \times 2 = 20 - 6 = 14$$ 4. **Step 3: Identify students using only one book** Since no student used Fountain alone, and Fountain is not used by any student alone, it implies Fountain is the book to replace. 5. **Step 4: Group marks and defend decision** Grouping students by book usage helps identify which book is least used alone (Fountain) and should be replaced to improve resource allocation. 6. **Step 5: Probability a student failed** Assuming failing marks are below 50 (common threshold), count marks below 50 from the given data: Marks below 50: 30, 26, 47, 49, 26, 43, 25, 45, 38, 44, 27, 46, 48, 32, 48, 32, 45, 40, 25, 45, 48, 45, 30, 38, 30, 28, 24, 48, 30, 28, 35, 35, 17, 9, 8, 2 Count = 36 Total students = 50 Probability student failed = $\frac{36}{50} = 0.72$ **Final answers:** - (a)(i) Grouping shows Fountain is unused alone and should be replaced. - (a)(ii) Statistical diagram can be a Venn diagram showing overlaps. - (b)(i) Fountain book should be replaced because no student reads it alone. - (b)(ii) Probability a student failed is $0.72$ (72%).