# In a School 150 students passed X Standard examination. 95 students applied for Group I and 82 students applied for Group II in the Higher Secondary course. If 20 students applied neither of the two, how many students applied for both groups?

1
by tamilcsea

2014-09-16T22:24:00+05:30

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Let A = set of students who applied for group I
B = set of students who applied for group II
U = set of students who passed X standard exam
A Λ B = set of students who applied for both groups
A υ B = set of students who applied for group I or group II
~ (A υ B) = set of students who did not apply for any group

U  = (A υ B) + ~(A υ B)

Cardinatlity of U = cardinality of ~(A υ B) + cardinality of (A υ B)
150 = 20 + cardinality (A υ B)

A υ B = 150-20 = 130
A υ B = A + B - A Λ B
Cardinality of (A υ B) = cardinality of A + cardinality of B - cardinality of (A Λ B)
130 = 95 + 82 - cardinality ( A Λ B)

Cardinality of A Λ B = 47

So 47 students applied for both groups.