Question

The report of one survey of 100 students studying languages Hindi,
Bengali and Tamil are as follows:

All the three languages 5 students, Hindi and Bengali 10 students,
Bengali and Tamil 8 students, Tamil and Hindi 12 students, Bengali
Idents. Hindi 23 students, Tamil 50 students. Find the number
of the students who are studying-
(i) only one language
(ii) none
(iii) at least two language

​

Answer 1

(i) only one language - 58 , (ii) none = 22 ,  (iii) at least two languages. = 20

Step-by-step explanation:

Assuming Bengali 30 students,  

H ∩ B ∩ T =  5

H ∩ B = 10

B ∩ T = 8

T ∩ H = 12

B = 30

H = 23

T = 50

Only B  = B - B ∩ T  - H ∩ B + H ∩ B ∩ T

= 30 - 8 - 10 + 5

= 17

only H = H - T ∩ H  - H ∩ B + H ∩ B ∩ T

= 23 - 12 - 10 + 5

= 6

Only T = T - T ∩ H  - T ∩ B + H ∩ B ∩ T

= 50 - 12 - 8 + 5

= 35

Only one language = 17 + 6 + 35 = 58

Total = B + H + T  - H ∩ B - B ∩ T - T ∩ H  + H ∩ B ∩ T + None

=> 100 = 30 + 23 + 50 - 10 - 8 - 12 + 5 + None

=> none = 22

atleast two language  = 100 - none - only one langugae

= 100 - 58 - 22

= 20

or   H ∩ B + B ∩ T + T ∩ H - 2H ∩ B ∩ T

= 10 + 8 + 12 - 2 * 5

= 20

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