Answer to Question #287785 in Discrete Mathematics for Sandy

Question

4. 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 ?

Answer 1

a)

$$N(E\cup H\cup M)=N(E)+N(H)+N(M)$$

$$-N(E\cap H)-N(E\cap M)-N(H\cap M)$$

$$+N(E\cap H\cap M)$$

$$=30+45+60-10-20-15+5=95$$

$$N((E\cup H\cup M)^C)=N(U)-N(E\cup H\cup M)$$

$$=100-95=5$$

5 students speak none of the three languages.

b)

$$N(\text{ exactly two languages})$$

$$=N(E\cap H)-N(E\cap H\cap M)$$

$$+N(E\cap M)-N(E\cap H\cap M)$$

$$+N(H\cap M)-N(E\cap H\cap M)$$

$$=10-5+20-5+15-5=30$$

30 students speak exactly two languages.