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 only comedy movie.

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 only comedy movie. Correct Answer 11

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, the number of people who liked only  comedy movie is given by say X.

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

⇒ X = 29 – 14 – 12 + 8 = 11

Hence, there are 11 people who liked only comedy movie.