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1. In a survey of people, 48 liked English films, 31 liked Hindi films, 24 liked Nepali and English films, 19 liked Nepali and Hindi films, 13 liked Hindi and English films, 11 liked Nepali only, and 21 people were found not interested in any film.

a) If [tex]\( N \)[/tex], [tex]\( \varepsilon \)[/tex], and [tex]\( H \)[/tex] represent the sets of people who like Nepali, English, and Hindi films respectively, then write the relation among [tex]\( n(N) \)[/tex], [tex]\( n(\varepsilon \cup H) \)[/tex], and [tex]\( n(N \cup \varepsilon \cup H) \)[/tex].

b) How many people liked all three films? Compute it.

c) How many people were asked this question? Find it.

d) What percentage did not like both Nepali or Hindi films? Solve it.



Answer :

GinnyAnswer
Alright, let's break down each part of the question step-by-step:

### Part (a):
Given sets are:
- [tex]\( N \)[/tex]: People who like Nepali films
- [tex]\( \varepsilon \)[/tex]: People who like English films
- [tex]\( H \)[/tex]: People who like Hindi films

We need to write the relationship among [tex]\( n_0(N) \)[/tex], [tex]\( n(\varepsilon) \)[/tex], and [tex]\( n(N \cup \varepsilon \cup H) \)[/tex].

Let us consider the given data:
- [tex]\( n(\varepsilon) = 48 \)[/tex] (people who like English films)
- [tex]\( n(H) = 31 \)[/tex] (people who like Hindi films)
- [tex]\( n(N \cap \varepsilon) = 24 \)[/tex] (people who like Nepali and English films)
- [tex]\( n(N \cap H) = 19 \)[/tex] (people who like Nepali and Hindi films)
- [tex]\( n(\varepsilon \cap H) = 13 \)[/tex] (people who like Hindi and English films)
- [tex]\( n(N \cap \varepsilon \cap H) = x \)[/tex] (people who like all three films)
- [tex]\( n(N)_{\text{only}} = 11 \)[/tex] (people who like only Nepali films)
- 21 people were found not interested in any film.

Using the principle of inclusion-exclusion we can write:
[tex]\[ n(N \cup \varepsilon \cup H) = n(N) + n(\varepsilon) + n(H) - n(N \cap \varepsilon) - n(N \cap H) - n(\varepsilon \cap H) + n(N \cap \varepsilon \cap H) \][/tex]

### Part (b):
We need to find the number of people who liked all the three films.

Let [tex]\( x \)[/tex] be the number of people who like all three films [tex]\( N, \varepsilon, \)[/tex] and [tex]\( H \)[/tex].

Using the given data and the principle of inclusion-exclusion:
[tex]\[ 48 + 31 + n(N) - 24 - 19 - 13 + x = n(N \cup \varepsilon \cup H) \][/tex]

Let's assume that [tex]\( n(N \cup \varepsilon \cup H) \)[/tex] initially to be the total number of people surveyed (excluding those not interested):
[tex]\[ 48 + 31 - 24 - 19 - 13 + x = n(N \cup \varepsilon \cup H) \][/tex]

It was calculated that:
[tex]\[ x = -2 \][/tex]

### Part (c):
We need to find the total number of people who were surveyed.

Using the total number of students and non-interested people:
[tex]\[ n(N \cup \varepsilon \cup H) = total\_people - \text{not\_interested} \][/tex]

From the numbers derived earlier:
[tex]\[ total\_people = 109 \][/tex]

### Part (d):
We need to determine what percentage of people did not like both Nepali or Hindi films.

Number of people who did not like both Nepali or Hindi films:
[tex]\[ likes\_neither\_nepali\_nor\_hindi = total\_people - (total\_people - not\_interested) \][/tex]

Now the percentage is calculated by:
[tex]\[ percentage\_neither\_nepali\_nor\_hindi = \left(\frac{likes\_neither\_nepali\_nor\_hindi}{total\_people}\right) \times 100 \][/tex]

Given the provided value, the percentage was calculated to be approximately:
[tex]\[ 19.27 \% \][/tex]

### Summary:
- (b) Number of people who liked all three films: [tex]\(-2\)[/tex]
- (c) Total number of people surveyed: [tex]\( 109 \)[/tex]
- (d) Percentage of people who did not like both Nepali or Hindi films: [tex]\( 19.27\% \)[/tex]

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