Question

In a class 82 people were asked their favorite fruits ?

39 likes apples, 50 liked bananas, 39 like pears, 21 liked apples and bananas, 18 liked bananas,

and pears, 19 liked apples and pears. How many liked all three types of fruits ? ​

Answer 1

$\textbf{Solution:}$

$\text{Let A,B and C denote set os students who like}$

$\text{Apples, Bananas and pears respectively}$

$\text{Then,}$

$\text{As per given data,}$

$n(A)=39$

$n(B)=50$

$n(C)=39$

$n(A{\cap}B)=21$

$n(B{\cap}C)=18$

$n(A{\cap}C)=19$

$\textbf{To find:}\;n(A{\cap}B{\cap}C)$

$\text{Using the formula,}$

$\bf\,n(A{\cap}B{\cap}C)=n(A)+n(B)+n(C)-n(A{\cap}B)-n(B{\cap}C)-n(A{\cap}C)+n(A{\cap}B{\cap}C)$

$\implies\,82=39+50+39-21-18-19+n(A{\cap}B{\cap}C)$

$\implies\,82=128-58+n(A{\cap}B{\cap}C)$

$\implies\,82=70+n(A{\cap}B{\cap}C)$

$\implies\,n(A{\cap}B{\cap}C)=82-70$

$\implies\,n(A{\cap}B{\cap}C)=12$

$\therefore\textbf{12 people like three type of fruits}$

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