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In a survey of 200 students of a school...

In a survey of 200 students of a school, it was found that 120 study Mathematics, 90 study Physics and 70 study Chemistry , 40 study Mathematics and Physics, 30 study Physics and Chemistry, 50 study Chemistry and Mathematics and 20 none of these subjects. Find the number of students who study all the three subjects.

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To solve the problem step by step, we will use the principle of inclusion-exclusion for three sets. ### Step 1: Define the sets Let: - \( M \) = number of students studying Mathematics = 120 - \( P \) = number of students studying Physics = 90 - \( C \) = number of students studying Chemistry = 70 - \( M \cap P \) = number of students studying both Mathematics and Physics = 40 - \( P \cap C \) = number of students studying both Physics and Chemistry = 30 - \( C \cap M \) = number of students studying both Chemistry and Mathematics = 50 - \( n \) = number of students studying none of these subjects = 20 ### Step 2: Calculate the total number of students studying at least one subject The total number of students surveyed is 200. Since 20 students study none of the subjects, the number of students studying at least one subject is: \[ n(M \cup P \cup C) = 200 - 20 = 180 \] ### Step 3: Apply the principle of inclusion-exclusion According to the principle of inclusion-exclusion for three sets, we have: \[ n(M \cup P \cup C) = n(M) + n(P) + n(C) - n(M \cap P) - n(P \cap C) - n(C \cap M) + n(M \cap P \cap C) \] Substituting the values we have: \[ 180 = 120 + 90 + 70 - 40 - 30 - 50 + n(M \cap P \cap C) \] ### Step 4: Simplify the equation Now, let's simplify the right side: \[ 180 = 120 + 90 + 70 - 40 - 30 - 50 + n(M \cap P \cap C) \] Calculating the sum: \[ 180 = 280 - 120 + n(M \cap P \cap C) \] \[ 180 = 160 + n(M \cap P \cap C) \] ### Step 5: Solve for \( n(M \cap P \cap C) \) Now, we can isolate \( n(M \cap P \cap C) \): \[ n(M \cap P \cap C) = 180 - 160 \] \[ n(M \cap P \cap C) = 20 \] ### Conclusion Thus, the number of students who study all three subjects (Mathematics, Physics, and Chemistry) is **20**. ---
To solve the problem step by step, we will use the principle of inclusion-exclusion for three sets. ### Step 1: Define the sets Let: - \( M \) = number of students studying Mathematics = 120 - \( P \) = number of students studying Physics = 90 - \( C \) = number of students studying Chemistry = 70 - \( M \cap P \) = number of students studying both Mathematics and Physics = 40 ...
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