A group of 40 students appeared in an examination of 3 subjects - Mathematics, Physics & Chemistry. It was found that all students passed in at least one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16 students passed in Chemistry, at most 11 students passed in both Mathematics and Physics, at most 15 students passed in both Physics and Chemistry, at most 15 students passed in bo

Step-by-step solution

Given, n M = 20 , n P = 25 , n C = 16 And n M ∩ P ≤ 11 , n M ∩ C ≤ 15 , n P ∩ C ≤ 15 Now, plotting the diagram and taking all three subjects passed as x we get, Now, for maximum let x = 11 . So, the venn diagram will be, Now, using the given conditions we get, 11 + z ≤ 15 ⇒ z ≤ 4 11 + y ≤ 15 ⇒ y ≤ 4 Now finding the sum of all we get, 9 - z + 0 + 14 - y + z + 11 + y + 5 - y - z = 40 ⇒ y + z = - 1 which is not possible hence the value of x = 11 is not possible, Now taking the maximum value as x = 10 , We get, 10 + z