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(A) Let $U$ be the set of consumers surveyed,$S$ be the set of consumers who like product $A$,and $T$ be the set of consumers who like product $B.$ Gi

Vedclass · Accepted Answer

$170$

Vedclass · Answer

(A) Let $U$ be the set of consumers surveyed,$S$ be the set of consumers who like product $A$,and $T$ be the set of consumers who like product $B.$ Given that $n(U) = 1000$,$n(S) = 720$,and $n(T) = 450.$ We know the formula for the union of two sets is $n(S \cup T) = n(S) + n(T) - n(S \cap T).$ Substituting the values,we get $n(S \cup T) = 720 + 450 - n(S \cap T) = 1170 - n(S \cap T).$ Since $S \cup T$ is a subset of $U$,the maximum value of $n(S \cup T)$ is $n(U) = 1000.$ Therefore,$1000 \geq 1170 - n(S \cap T).$ Rearranging the inequality,we get $n(S \cap T) \geq 1170 - 1000 = 170.$ Thus,the least number of consumers who liked both products is $170.$

$A$ market research group conducted a survey of $1000$ consumers and reported that $720$ consumers like product $A$ and $450$ consumers like product $B$. What is the least number of consumers that must have liked both products?

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