Question

A market research group conducted a survey of 2000 consumers and reported that 1720 consumers like product P₁ and 1450 consumers like product P₂. What is the least number that must have liked both the products?

Options

• 1150

• 2000

• 1170

• 2500

Answer 1

1170

Explanation:

Let U be the set of all consumers who were questioned, A be the set of consumers who liked product P₁ and B be the set of consumers who liked product P₂.

It is given that n(U) = 2000, n(A) = 1720, n(B) = 1450.

n(A ∪ B) = n(A) + n(B) – n(A n B)

n(A ∪ B) = 1720 + 1450 – n(A ∩ B) = 3170 – n(A ∩ B)

Since, A ∪ B ⊆ U

∴ n(A ∪ B) ≤ n(U)

=> 3170 – n(A ∩ B) ≤ 2000

=> 3170 – 2000 ≤ n(A ∩ B)

=> n(A ∩ B) ≥ 1170

Thus, the least value of n(A ∩ B) is 1170.

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