In a school of 4000 students, 2000 know French, 3000 know Tamil and 500 know Hindi, 1500 know French and Tamil, 300 know French and Hindi, 200 know Tamil and Hindi and 50 know all the three languages.

(i) How many do not know any of the three languages?

(ii) How many know at least one language?

(iii) How many know only two languages?

Let total student be U and students who know Tamil, Hindi and French be T, H, F respectively.

We know that,

No of student, U = 4000

No students who know Tamil = T = 3000

No students who know French = F = 2000

No students who know Hindi = H = 500

No students who know French and Tamil = (F ⋂ T) = 1500

No students who know Tamil and Hindi = (T ⋂ H) = 200

No students who know Hindi and French = (H ⋂ F) = 300

No students who know Tamil, Hindi and french = (T ⋂ H ⋂ F) = 50

it can be represented in a venn diagram

Student who knows only French and Tamil,

(T⋂F⋂H’) = [(T⋂F) – (T⋂H⋂F)] = 1500 – 50 = 1450

Student who knows only Hindi and Tamil,

(T⋂H⋂F’) = (T⋂H) – (T⋂H⋂F) = 200 – 50 = 150

Student who knows only French and Hindi,

(H⋂F⋂T’) = (F⋂H) – (T⋂H⋂F) = 300 – 50 = 250

Student who knows only French, F’ = F – (H⋂F⋂T’) – (T⋂H⋂F) – (T⋂F⋂H’)

= 2000 – 1450 – 250 – 50 = 250

Student who knows only Tamil, T’ = T – (T⋂H⋂F’) – (T⋂F⋂H’) – (T⋂H⋂F)

= 3000 – 150 – 1450 – 50 = 1350

Student who knows only Hindi, H’ = H – (T⋂H⋂F’) – (T⋂H⋂F) – (H⋂F⋂T’)

= 500 – 150 – 50 – 250 = 50

(i) So no of student who don’t any of the three languages are the students who lies outside the set of T, H, F denoted as (F ⋃ H ⋃ T)’.

from cardinality of sets we know that

n(F U H U T) = n(F) + n(H) + n(T) – n(F ∩ H) – n(H ∩ T) – n(F ∩ T) + n(F ∩ H ∩ T)

putting the values

n(F ⋃ H ⋃ T) = 2000 + 500 + 3000 – 300 – 200 – 1500 + 50

= 3550

So no of student who don’t any of the three languages (F ⋃ H ⋃ T)’ given as

n(F U H U T)’ = n(U) – n(F U H U T)

= 4000 – 3550

= 450

Hence 450 no of student who don’t any of the three languages.

(ii) At least one language means the student must know minimum of one language while at max it can be 3. So its clear that here we have to find the total number of student who knows the language which is denoted as n(F⋃ H ⋃ T)

Using cardinality of set theory

n(F U H U T) = n(F) + n(H) + n(T) – n(F ∩ H) – n(H ∩ T) – n(F ∩ T) + n(F ∩ H ∩ T)

putting the values

n(F ⋃ H ⋃ T) = 2000 + 500 + 3000 – 300 – 200 – 1500 + 50

= 3550

So 3550 number of student knows at least one language.

(iii) Here we have to find the number of student who knows only two language.

So

Student who knows only two language =

= (T⋂F⋂H’) + (T⋂H⋂F’) + (H⋂F⋂T’)

= 1450 + 150 + 250

= 1850

So 1850 number student knows two languages only.