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Sagot :
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]
### 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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