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. $50$
• B. $60$
• C. $45$
• D. $55$

Answer 1

The correct option is D $55$

Let, set of students who study :
English→E, French→F, Spanish→S

Given: $n\left(E\right)=30,n\left(F\right)=25,n\left(S\right)=20$
$n(E\cap F)=8,n(F\cap S)=5,n(S\cap E)=7$
$n(E\cap F\cap S)=0$
To find : Number of students who study at least one of the three languages $=n\left(EUFUS\right)$
We know the formula,
$n\left(AUBUC\right)=n\left(A\right)+n\left(B\right)+n\left(C\right)-n(A\cap B)-n(B\cap C)-n(C\cap A)+n(A\cap B\cap C)$

so, $n\left(EUFUS\right)=n\left(E\right)+\left(F\right)+n\left(S\right)-n(E\cap F)-n(F\cap S)-n(S\cap A)+n(E\cap F\cap S)$
$n\left(EUFUS\right)=30+25+20-8-5-7-0$
$=55$