A survey of 100 students produced the following information: 30 speak English, 45 speak Hindi and 60 speak Marathi. It is found that 10 speak both English and Hindi, 20 speak both Hindi and Marathi, 15 speak both English and Marathi and 5 speak all the three languages. a) How many of these students speak none of the three languages ? b) How many of these students speak exactly two languages ?

Solution:

Let's denote English, Hindi and Marathi by E, H and M respectively.

30 speak English, $n(E)=30-10-5-5=10$

45 speak Hindi, $n(H)=45-15-5-5=20$

60 speak Marathi, $n(M)=60-15-10-5=30$

10 speak both English and Hindi, $n(E\cap H)=10-5=5$

20 speak both Hindi and Marathi, $n( H\cap M)=20-5=15$

15 speak both English and Marathi, $n(E\cap M)=15-5=10$

5 speak all the three languages, $n(E\cap H\cap M)=5$

a) How many of these students speak none of the three languages ?

Ans: From venn diagram, there are 5 students speak none of the three languages.

Or the students speak none of the three languages = 100 - (10+5+20+10+5+15+30)

= 100-95

=5

b) How many of these students speak exactly two languages ?

Ans: The students speak exactly two language = 5+10+15=30