Question

A market research group conducted a survey of $2000$ consumers and reported that $1720$ consumers like product $P_1$ and $1450$ consumers like product $P_2$. What is the least number that must have liked both the products?

• A. $1150$
• B. $2000$
• C. $1170$
• D. $2500$

Answer 1

Answer: (C)

Let $U$ be the set of all consumers who were questioned, $A$ be the set of consumers who liked product $P_1$ and $B$ be the set of consumers who liked the product $P_2$.
It is given that $n(U) =2000$, $n(A) = 1720$, $n(B) = 1450$.
$n(A \cup B) = n(A) + n(B) - n(A \cap B)$
$n(A \cup B) = 1720 + 1450 - n(A \cap B) = 3170 - n(A \cap B)$
Since, $A \cup B \subseteq U$
$\therefore n(A \cup B ) \le n(U)$
$\Rightarrow 3170 - n(A \cap B) \le 2000$
$\Rightarrow 3170-2000 \le n (A \cap B)$
$\Rightarrow n(A \cap B) \ge 1170$
Thus, the least value of $n(A \cap B)$ is $1170$.
Hence, the least number of consumers who liked both the products is $1170$.