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Question

In a school three languages English, French and Spanish are taught. 30 students study English, 25 study French and 20 study Spanish. Although no student studies all three languages, 8 students study both English and French, 5 students study both French and Spanish and 7 students study both Spanish and English. How many students study at least one of the three languages?

A
50
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B
60
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C
45
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D
55
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Solution

The correct option is D 55
Let, set of students who study :
English→E, French→F, Spanish→S

Given: n(E)=30,n(F)=25,n(S)=20
n(EF)=8,n(FS)=5,n(SE)=7
n(EFS)=0
To find : Number of students who study at least one of the three languages =n(EUFUS)
We know the formula,
n(AUBUC)=n(A)+n(B)+n(C)n(AB)n(BC)n(CA)+n(ABC)

so, n(EUFUS)=n(E)+(F)+n(S)n(EF)n(FS)n(SA)+n(EFS)
n(EUFUS)=30+25+208570
=55


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n(A∪B∪C) = n(A) + n(B) + n(C) − n(A∩B) − n(B∩C) − n(C∩A) + n(A∩B∩C)
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