Question

Suppose 100 out of 120 study French, German and Russian. It is given that 65 study

French, 45 study German, 42 study Russian, 20 study French and German, 25 Study

French and Russian and 15 students study German and Russian. Find the number of

students who study all the three languages.​

Answer 1

Given:

Total number of students = 120

Total number of students that study French, German and Russian n(F∪G∪R)= 100

Number of students that study French n(F) = 65

Number of students that study German n(G) = 45

Number of students that study Russian n(R) = 42

Number of students that study French and German n(F∩G) = 20

Number of students that study French and Russian n(F∩R) = 25

Number of students that study German and Russian n(G∩R) = 15

To find:

The number of  students who study all the three languages.​

Solution:

n(F∪G∪R) = n(F) + n(G) + n(R) - n(F∩G) - n(G∩R) -n(R∩F) + n(F∩G∩R)

100 = 65 +45 + 42 - 20 - 25 - 15 + n(F∩G∩R)

100 = 92 + n(F∩G∩R)

n(F∩G∩R) = 100 - 92

n(F∩G∩R) = 8

The number of  students who study all the three languages is 8.