Question

discretemathematics- in a class of 80 students,50 students konw english,55 know french,46 know german language,37 know english and french,28 student know french and german,25 student know english and german,7 students know none of the language,then i)how many student know all the 3 subjects. ii)how many students know exactly 2 languages? iii)how many know only one languages? using |AUBUC|=|A|+|B|+|C|-|A^B|-|A^C|-|B^C|+|A^B^C|

Answer 1

Answer:

12

Step-by-step explanation:

all students 80-7 =73

Answer 2

The number of students who know all the 3 subjects is 12; the number of students who know exactly 2 languages is 54, and; the number of students who know only one of the languages is 7.

Given:

Total number of students = 80

Number of students who know English = 50

Number of students who know French = 55

Number of students who know German = 46

Number of students who know English and French = 37

Number of students who know French and German = 28

Number of students who know English and German = 25

Number of students who know none of the subjects = 7

To Find:

1. Number of students who know all the 3 subjects

2. Number of students who know exactly 2 languages

3. Number of students who know only one of the languages

Solution:

Let a be the number of students who know all the 3 languages.

The given information can be represented in the form of a Venn diagram as shown in the figure below.

Now, the total number of students = number of students who know English + French + German + students who know none of the 3

⇒ 80 = 50 + (55 - 28 + a - 37) + (46 - 28 + a - 25) + 28 - a + 7

80 = 57 - 10 + a - 7 + a + 28 - a

80 = 68 + a

a = 12

Now, the number of people who know exactly 2 languages = 37 - a + 28 - a + 25 - a

= 90 - 3a

= 90 - 36

= 54

Lastly, the number of people who know exactly 1 language = (55 - 28 + a - 37) + (46 - 28 + a - 25) + (50 - 37 - 25 + a)

= a - 10 + a - 7 + a - 12

= 3a - 29

= 7

Thus, the number of students who know all the 3 subjects is 12; the number of students who know exactly 2 languages is 54, and; the number of students who know only one of the languages is 7.

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