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.
ScanSolve AI Answer
Right master, let's go. Step 1: Identify the given information. Let U be the universal set of students. Let M be the set of students taking Mathematics. Let P be the set of students taking Physics. Let C be the set of students taking Chemistry. We are given: Total number of students, |U| = 120 Number of students taking Mathematics, |M| = 70 Number of students taking Physics, |P| = 65 Number of students taking Chemistry, |C| = 50 Number of students taking Mathematics and Physics, |M P| = 30 Number of students taking Mathematics and Chemistry, |M C| = 25 Number of students taking Physics and Chemistry, |P C| = 20 Number of students taking all three subjects, |M P C| = 10 Step 2: Calculate the number of students in each disjoint region of the Venn diagram. • Students taking exactly two subjects: |M P only| = |M P| - |M P C| = 30 - 10 = 20 |M C only| = |M C| - |M P C| = 25 - 10 = 15 |P C only| = |P C| - |M P C| = 20 - 10 = 10 • Students taking exactly one subject: |M only| = |M| - (|M P only| + |M C only| + |M P C|) |M only| = 70 - (20 + 15 + 10) = 70 - 45 = 25 |P only| = |P| - (|M P only| + |P C only| + |M P C|) |P only| = 65 - (20 + 10 + 10) = 65 - 40 = 25 |C only| = |C| - (|M C only| + |P C only| + |M P C|) |C only| = 50 - (15 + 10 + 10) = 50 - 35 = 15 Step 3: Answer part (a). a) How many students take only mathematics? The number of students who take only Mathematics is |M only|. |M only| = 25 The number of students who take only mathematics is 25. Step 4: Answer part (b). b) How many take exactly two subjects? The number of students who take exactly two subjects is the sum of those taking (M and P only), (M and C only), and (P and C only). Exactly two subjects = |M P only| + |M C only| + |P C only| Exactly two subjects = 20 + 15 + 10 = 45 The number of students who take exactly two subjects is 45. Step 5: Answer part (c). c) How many students take none of the three subjects? First, calculate the total number of students taking at least one subject: |M P C| = |M only| + |P only| + |C only| + |M P only| + |M C only| + |P C only| + |M P C| |M P C| = 25 + 25 + 15 + 20 + 15 + 10 + 10 = 120 The number of students taking none of the three subjects is |U| - |M P C|. None of the subjects = 120 - 120 = 0 The number of students who take none of the three subjects is 0. Step 6: Answer part (d). d) Draw a Venn diagram with all the information. As an AI, I cannot physically draw a diagram. However, I can describe the Venn diagram by listing the number of students in each distinct region: • Students taking only Mathematics: 25 • Students taking only Physics: 25 • Students taking only Chemistry: 15 • Students taking Mathematics and Physics only (not Chemistry): 20 • Students taking Mathematics and Chemistry only (not Physics): 15 • Students taking Physics and Chemistry only (not Mathematics): 10 • Students taking Mathematics, Physics, and Chemistry: 10 • Students taking none of the subjects: 0 Send me the next one 📸