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
Let M be the set of students taking Mathematics, A be the set of students taking Agriculture, and C be the set of students taking Chichewa. The total number of students is N = 40. Given information: |M| = 19 |A| = 13 |C| = 17 Number of students taking none of the subjects = 9 |M A| = 8 |A C| = 7 |C M| = 9 Step 1: Calculate the number of students taking at least one subject. The number of students taking at least one subject is the total number of students minus those taking none. |M A C| = N - (students taking none) |M A C| = 40 - 9 |M A C| = 31 Step 2: Apply the Principle of Inclusion-Exclusion for three sets. The formula is: |M A C| = |M| + |A| + |C| - (|M A| + |A C| + |C M|) + |M A C| Substitute the known values into the formula: 31 = 19 + 13 + 17 - (8 + 7 + 9) + |M A C| 31 = 49 - 24 + |M A C| 31 = 25 + |M A C| Step 3: Solve for the number of students taking all three subjects. |M A C| = 31 - 25 |M A C| = 6 So, 6 students take all three subjects. Step 4: Calculate the number of students in each specific region for the Venn diagram. Let x = |M A C| = 6. • Students taking only Mathematics and Agriculture: |M A| - x = 8 - 6 = 2 • Students taking only Agriculture and Chichewa: |A C| - x = 7 - 6 = 1 • Students taking only Chichewa and Mathematics: |C M| - x = 9 - 6 = 3 • Students taking only Mathematics: |M| - (only M and A) - (only C and M) - x = 19 - 2 - 3 - 6 = 8 • Students taking only Agriculture: |A| - (only M and A) - (only A and C) - x = 13 - 2 - 1 - 6 = 4 • Students taking only Chichewa: |C| - (only A and C) - (only C and M) - x = 17 - 1 - 3 - 6 = 7 Step 5: Describe the Venn diagram. A Venn diagram would consist of a rectangle representing the universal set of 40 students. Inside this rectangle, there would be three overlapping circles labeled M (Mathematics), A (Agriculture), and C (Chichewa). The numbers in each region would be: • The intersection of all three circles (M A C): 6 students. • The region for M A only (not including C): 2 students. • The region for A C only (not including M): 1 student. • The region for C M only (not including A): 3 students. • The region for M only (not A or C): 8 students. • The region for A only (not M or C): 4 students. • The region for C only (not M or A): 7 students. • The region outside all three circles (students taking none): 9 students. To verify, sum all regions: 6 + 2 + 1 + 3 + 8 + 4 + 7 + 9 = 40. This matches the total number of students. The number of students who take all three subjects is 6 students. Send me the next one 📸