Question

A survey of 100 students was taken concerning their knowledge of foreign languages. The results were as follows: 30 know French19 know German16 know Russian11 know both French and Russian12 know both French and German6 know both German and Russian4 know all three languages How many students do NOT know any of the three languages?

Answer 1

Answer: There are 48 such students who did not know any of the three languages.

Step-by-step explanation:

Since we have given that

Number of students were taken for survey n(U)= 100

Number of students who know French n(F) = 30

Number of students who know German n(G) = 19

Number of students who know Russian n(R) = 16

Number of students who know both French and Russian n(F∩ R) = 11

Number of students who know both German and French n(G ∩ F) = 12

Number of students who know both Russian and German n(R ∩ G) = 6

Number of students who know all three languages = 4

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

n(F∪G∪R)=30+19+16-12-11-6+4

n(F∪G∪R)=52

So, the number of students who do not know any of the three languages is

$n(F\cap G\cap R)'=n(U)-n(F\cap G\cap R)\\\\n(F\cap G\cap R)'=100-52\\\\n(F\cap G\cap R)'=48$

Hence, there are 48 such students who did not know any of the three languages.

Answer 2

35 students would not know any of the three languages, because if 30 know French, 19 knew German, and 16 knew Russian, that must mean that 65 students know at least one of the three languages. So you subtract 65 from 100 and end up with 35