Question

In a survey of brand preference for toothpastes, $82$ of the population (number of people covered for the survey is $100$) liked at least one of the brands: I, II and III. $40$ of those liked brand I, $25$ liked brand II and $35$ liked brand III. If $8$ of those asked, showed liking for all the three brands, then what percentage of those liked more than one of the three brands?

• A. $13$
• B. $10$
• C. $8$
• D. $5$

Answer 1

Answer: (B)

$10$

By set theory

$n(I\cup II\cup III)=n\left(I\right)+n\left(II\right)+n\left(III\right)-n(I\cap II)-n(II\cap III)-n(I\cap III)+n(I\cap II\cap III)$

$82=40+25+35-x+8$

$\Rightarrow x=26$

But

$n(I\cap II)+n(II\cap III)+n(I\cap III)$ counts $n(I\cap II\cap III)$ thrice whereas it needs to be counted only once.

Hence answer $=26-2\times 8=10$