Question

A survey was carried out to find out the types of shampoo that a group of $150$ women have tried. It was found that $84$ women have used brand $A$ shampoo, $93$ have used brand $B$, and $69$ have used brand $C$ of these women, $45$ have tried brands $A$ and $B$, $25$ have tried brands $A$ and $C$ and $40$ have tried brand $B$ and $C$. Determine the number of women who have tried only brand $A$.

Answer 1

$n(A\cup B\cup C)=150$

$n\left(A\right)=84$

$n\left(B\right)=93$

$n\left(C\right)=69$

$n(A\cap C)=25$

$n(A\cap B)=45$

$n(B\cap C)=40$

a.

$n(A\cap B\cap C)=n(A\cup B\cup C)-n\left(A\right)-n\left(B\right)-n\left(C\right)+n(A\cap B)+n(A\cap C)-n(B\cap C)$

$=150-84-93-69+45+25+40=14$

b.

Number of women who only used brand A= $n\left(A\right)-n(A\cap B)-n(A\cap C)=84-45-25=14$