In an examination, there are three subjects ‘A’, ‘B’ and ‘C’. A student has to pass in each subject. 20% students failed in ‘A’, 22% students failed in ‘B’ and 16% students failed in ‘C’. The total number of students passing the whole examination lies between :
42% and 84%
42% and 78%
58% and 78%
58% and 84%
Answer is Wrong!
Answer is Right!
This question was previously asked in
UPSC CAPF – 2015
F(A) = 20%, P(A) = 100 – 20 = 80%
F(B) = 22%, P(B) = 100 – 22 = 78%
F(C) = 16%, P(C) = 100 – 16 = 84%
A student passes the whole examination if they pass in A AND B AND C. The percentage passing all is P(A ∩ B ∩ C).
Maximum percentage passing all: This occurs when the sets of students passing each subject overlap as much as possible. The maximum overlap is limited by the smallest passing percentage. So, max P(A ∩ B ∩ C) <= min(P(A), P(B), P(C)) = min(80, 78, 84) = 78%. Minimum percentage passing all: This occurs when the sets of students failing at least one subject (F(A U B U C)) are maximized. The maximum percentage failing at least one subject occurs when the failure sets are disjoint (no overlap). Max P(F(A U B U C)) <= F(A) + F(B) + F(C) = 20 + 22 + 16 = 58%. The minimum percentage passing all = 100% - Max P(F(A U B U C)) = 100% - 58% = 42%. Alternatively, using the inclusion-exclusion principle for passing percentages, minimum P(A ∩ B ∩ C) >= P(A) + P(B) + P(C) – 2 * 100% = 80 + 78 + 84 – 200 = 242 – 200 = 42%.
Thus, the total number of students passing the whole examination lies between 42% and 78%.