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In a set of 50 students, 28 passed in English, 30 in Hindi, and 12 passed in both. How many failed in both subjects?
4
6
8
10
4
Calculate the union of students passing at least one subject by adding individual passes and subtracting the intersection (28+30тИТ12=46), then subtract this result from the total number of students (50тИТ46=4).
Total students = 50, Passed in English = 28, Passed in Hindi = 30, Passed in both = 12
n(EтИкH)=n(E)+n(H)тИТn(EтИйH)
Calculate the union of students passing at least one subject by adding individual passes and subtracting the intersection (28+30тИТ12=46), then subtract this result from the total number of students (50тИТ46=4).
Many students subtract only the number of students who passed in one subject from the total, forgetting that the inclusion-exclusion principle requires accounting for the overlap of those who passed in both subjects.
Define sets
Let E be the set of students who passed in English and H be the set of students who passed in Hindi. We are given n(E)=28, n(H)=30, and n(EтИйH)=12.
n(E)=28,n(H)=30,n(EтИйH)=12
Find number of students who passed at least one subject
Using the Inclusion-Exclusion Principle, find the number of students who passed in English or Hindi or both: n(EтИкH)=28+30тИТ12=46.
n(EтИкH)=28+30тИТ12=46
Calculate failed students
The number of students who failed in both subjects is the total number of students minus those who passed at least one subject: 50тИТ46=4.
50тИТ46=4
A is correct because the number of students who passed in at least one subject is 46, leaving 4 students who failed in both subjects.
This logic is identical to finding the union of two events in Probability. Always identify the intersection first to avoid double-counting.