In a survey it was found that 21 people liked fictional movies, 26 liked horror movies and 29 people liked comedy movies. If 14 people liked fictional and horror movies, 12 people liked fictional and comedy movies, 14 people liked horror and comedy movies and 8 people liked all three kinds of movie. Find how many people liked exactly 2 type of movies.

In a survey it was found that 21 people liked fictional movies, 26 liked horror movies and 29 people liked comedy movies. If 14 people liked fictional and horror movies, 12 people liked fictional and comedy movies, 14 people liked horror and comedy movies and 8 people liked all three kinds of movie. Find how many people liked exactly 2 type of movies. Correct Answer 16

Concept:

Let A, B and C be three finite sets and U is the finite universal set, then

1. n (A ∪ B) = n (A) + n (B) – n (A ∩ B)
2. n (A ∪ B) = n (A) + n (B) ⇔ A ∩ B = ϕ
3. n (A - B) = n (A) – n (A ∩ B) = n (A ∩ B’)
4. n (A ∪ B ∪ C) = n (A) + n (B) + n (C) – n(A ∩ B) – n (B ∩ C) – n (A ∩ C) + n (A ∩ B ∩ C)
5. n (A’ ∪ B’) = n = n (U) – n (A ∩ B)
6. n (A’ ∩ B’) = n = n (U) – n (A ∪ B)
7. n (A Δ B) = n (A) + n (B) – 2 n (A ∩ B)
8. n (A’) = n (U) – n (A)

Calculation:

Let, F = No. of people who like fictional movies, H = No. of people who like horror movies and C = No. of people who like comedy movies.

Given: n (F) = 21, n (H) = 26, n (C) = 29, n (F ∩ H) = 14, n (F ∩ C) = 12, n (H ∩ C) = 14 and n (F ∩ H ∩ C) = 8.

So, number of people who liked exactly 2 type of movies is given by say X.

X = n (F ∩ H) + n (F ∩ C) + n (H ∩ C) – 3 × n (F ∩ H ∩ C)

⇒ X = 14 + 12 + 14 – 3 × 8 = 40 – 24 = 16.

Hence, there were 16 people who liked exactly 2 type of movies.