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

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 product $P_2$. It is given that

$n (U)= 2000, n (A) = 1720$ and $n (B) = 1450$

$∴n(A∪B) =n (A) + n (B) -n (A∩B)$

$⇒n (A∪B) = 1720 + 1450 -n (A∩B)$

$⇒n (A∪B) = 3170 -n (A∩B)$

Now,

$A∪B⊂U$

$⇒n (A∪B)≤n(U)$

$⇒3170-n (A∩B)≤2000$

$n (A∩B)≥1170$

Hence, the least number of consumers who like both the products is 1170.