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 languages.
Answers
(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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