In a school, a student can opt the language subject out of English, Hindi and Sanskrit either single

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19 Jul 2022 Permutation & Combination

In a school, a student can opt the language subject out of English, Hindi and Sanskrit either single or in a combination. 50 students opted English, 62 Hindi and 42 Sanskrit. If there are total 78 students and only 12 opted for all three languages, then the number of students who opted for exactly two languages is

OPtion

1) 72
2) 62
3) 76
4) 52
5) 45
6) 80
7) 50
8) 48
9) 60
10) None of these

Solution

Let E, H, S be the sets of students who opted for English, Hindi and Sanskrit respectively.
n(E)=50
n(H)=62
n(S)=42
n(E∪H∪S)=78
n(F∩B∩C)=12
We know, n(E∪H∪S)=n(E)+n(H)+n(S)−n(E∩H)−n(H∩S)−n(E∩S)+n(E∩H∩S)
78 = 50 + 62 + 42 −n(E∩H)−n(H∩S)−n(E∩S) + 12
⇒ n(E∩H) + n(H∩S) + n(E∩S) = 88
"The number of students who opted for exactly two languages = n(E∩H) + n(H∩S) + n(E∩S) − 3×n(E∩H∩S) = 88 − 3x12 = 52

Correct Option: 4 Best Solution (1)

G.S.Kantharao   1 year AGO

n(E)=50,n(H)=62,n(S)=42
n(EUHUS)=78
n(E^H^S)=12
Opted strictly two only=
n(E)+n(H)+n(S)-n(EUHUS)-2n(E^H^S)
=50+62+42-78-2*12
=154-102=52
My option is 4👈

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