This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Welcome back — been a while! Let's pick up where you left off. Let S be the set of students who had Sculpture books. Let G be the set of students who had Graphics books. Let L be the set of students who had Leather work books. Given information: Total students in class = 65 Students absent = 2 Number of students who had at least one book, n(S G L) = 65 - 2 = 63. n(S) = 28 n(G) = 30 n(L) = 32 n(S G) = 11 n(S L) = 12 n(G L) = 9 a) Find the number of students who had all three books. Step 1: Use the Principle of Inclusion-Exclusion for three sets. The formula is: n(S G L) = n(S) + n(G) + n(L) - n(S G) - n(S L) - n(G L) + n(S G L) Substitute the given values: 63 = 28 + 30 + 32 - 11 - 12 - 9 + n(S G L) 63 = 90 - 32 + n(S G L) 63 = 58 + n(S G L) Solve for n(S G L): n(S G L) = 63 - 58 n(S G L) = 5 The number of students who had all three books is 5. b) Find the number of students who had exactly two books. Step 2: Calculate the number of students who had exactly two specific books. Number of students with Sculpture and Graphics books ONLY: n(S G only) = n(S G) - n(S G L) = 11 - 5 = 6 Number of students with Sculpture and Leather books ONLY: n(S L only) = n(S L) - n(S G L) = 12 - 5 = 7 Number of students with Graphics and Leather books ONLY: n(G L only) = n(G L) - n(S G L) = 9 - 5 = 4 The number of students who had exactly two books is the sum of these values: 6 + 7 + 4 = 17 The number of students who had exactly two books is 17. c) Find the number of students who had only one book. Step 3: Calculate the number of students who had only one specific book. Number of students with Sculpture books ONLY: n(S only) = n(S) - n(S G only) - n(S L only) - n(S G L) n(S only) = 28 - 6 - 7 - 5 = 28 - 18 = 10 Number of students with Graphics books ONLY: n(G only) = n(G) - n(S G only) - n(G L only) - n(S G L) n(G only) = 30 - 6 - 4 - 5 = 30 - 15 = 15 Number of students with Leather books ONLY: n(L only) = n(L) - n(S L only) - n(G L only) - n(S G L) n(L only) = 32 - 7 - 4 - 5 = 32 - 16 = 16 The number of students who had only one book is the sum of these values: 10 + 15 + 16 = 41 The number of students who had only one book is 41. Send me the next one 📸