A market research group conducted a survey of 2000 consumers and reported that 1720 consumers liked product `P_(1)and 1450` consumers liked products `P_(2)` the least number of consumers who must have liked both the products is

Options

A. 1170
B. 3170
C. 270
D. none of these

Answer

A

Explanation

Let U be the set of all consumers who were questioned ,A be the set consumers who liked product `p_(2) ` it is given that
`n(U) = 2000,n (A) =1720 and n(B) = 1450`
`therefore n(Acup B) =n(A) +n(B) -n(A cap B)`
`implies n(Acup B) = 3170 -N(AcapB)`
now ,
`A cup B subU`
`implies n(Acap B)le n (U)`
`implies 3170 -n(A cap B)le 2000`
`implies n(Acap B)ge 1170.`
hence the least number of consumers who like both the products is 1170.