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
Welcome back Harriet — been a while! Let's pick up where you left off. Let M represent students studying Mathematics, P for Physics, and C for Chemistry. We are given the following information: Total students |U| = 200 |M| = 120 |P| = 90 |C| = 70 |M P| = 40 |P C| = 30 |C M| = 50 Number of students studying none of these subjects |M P C|' = 20 Step 1: Find the number of students who study at least one of the three subjects. The number of students studying at least one subject is the total number of students minus those who study none. |M P C| = |U| - |M P C|' |M P C| = 200 - 20 |M P C| = 180 Step 2: Use the Principle of Inclusion-Exclusion to find the number of students who study all three subjects. The formula for the union of three sets is: |M P C| = |M| + |P| + |C| - |M P| - |P C| - |C M| + |M P C| Substitute the known values: 180 = 120 + 90 + 70 - 40 - 30 - 50 + |M P C| 180 = 280 - 120 + |M P C| 180 = 160 + |M P C| |M P C| = 180 - 160 |M P C| = 20 a) The number of students who study all three subjects is 20 students. Step 3: Calculate the number of students studying exactly two subjects. These values will be used to find the number of students studying only one subject. |M P only| = |M P| - |M P C| = 40 - 20 = 20 |P C only| = |P C| - |M P C| = 30 - 20 = 10 |C M only| = |C M| - |M P C| = 50 - 20 = 30 Step 4: Calculate the number of students who study only Physics. To find students who study only Physics, subtract those who study Physics with Mathematics (only), Physics with Chemistry (only), and all three subjects from the total number of Physics students. |P only| = |P| - |M P only| - |P C only| - |M P C| |P only| = 90 - 20 - 10 - 20 |P only| = 90 - 50 |P only| = 40 b) (i) The number of students who study Physics only is 40 students. Step 5: Calculate the number of students who study only Chemistry. Similarly, for Chemistry only: |C only| = |C| - |C M only| - |P C only| - |M P C| |C only| = 70 - 30 - 10 - 20 |C only| = 70 - 60 |C only| = 10 b) (ii) The number of students who study Chemistry only is 10 students. Drop the next question! 📸