To solve the problem of how many students offer both Mathematics and Physics, we can use the principle of inclusion-exclusion. Let's break down the problem step-by-step. ### Step 1: Understand the Given Information - Total number of students in the class: **40** - Students offering Mathematics: **32** - Students offering Physics: **24** - Students offering neither Mathematics nor Physics: **4** ### Step 2: Calculate Students Offering Either Subject First, we need to find out how many students are offering at least one of the subjects (Mathematics or Physics). Since 4 students offer neither subject, we can calculate the number of students who offer at least one subject as follows: \[ \text{Students offering at least one subject} = \text{Total students} - \text{Students offering neither} \] \[ \text{Students offering at least one subject} = 40 - 4 = 36 \] ### Step 3: Use the Inclusion-Exclusion Principle The inclusion-exclusion principle states that: \[ \text{Number of students offering at least one subject} = \text{Number of students offering Mathematics} + \text{Number of students offering Physics} - \text{Number of students offering both subjects} \] Let \( x \) be the number of students offering both Mathematics and Physics. We can set up the equation: \[ 36 = 32 + 24 - x \] ### Step 4: Solve for \( x \) Now, we can simplify the equation: \[ 36 = 56 - x \] Rearranging gives us: \[ x = 56 - 36 \] \[ x = 20 \] ### Conclusion Thus, the number of students who offer both Mathematics and Physics is **20**. ### Explanation of Other Options - **Option A (20)**: This is the correct answer as derived from our calculations. - **Option B (16)**: This option is incorrect because it does not account for the total number of students correctly when applying the inclusion-exclusion principle. - **Option C (8)**: This option is also incorrect as it underestimates the overlap between the two subjects. - **Option D (4)**: This option is incorrect as it suggests a very small overlap, which does not align with the total number of students offering either subject. ### Revision Summary - Use the inclusion-exclusion principle to find the number of students in overlapping sets. - Calculate the total number of students offering at least one subject by subtracting those offering neither. - Set up the equation based on the principle and solve for the unknown. - Always double-check your calculations to ensure accuracy. This thorough approach ensures that you understand not just the answer, but the reasoning behind it, which is crucial for success in mathematics exams.