Question

8. 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?

Answer 1

CARDINAL NUMBER OF A FINITE SET :
The Cardinal number of a finite set A is the number of distinct elements in the set A. It is denoted by n(A).
•It is not possible to define the Cardinal number of an infinite set.
•The Cardinal number of the empty set is zero.
•The Cardinal number of a singleton set is 1.

Intersection of two sets :
The intersection of the sets a and b is the set of all the elements which belong to both A and B. It is denoted by A ∩ B (“ A intersection B”).
•If A and B do not have any element in common then A ∩ B= a null set = Ø.

SOLUTION :
Let U = total students, F= students who know French, T =  students who know Tamil, H= students who know Hindi, n( F ∩ T)= number of students who know French & Tamil, n( F ∩ H)= number of students who know French & Hindi, n( T ∩ H)= number of students who know Tamil & Hindi, n( T ∩ H ∩ F)= number of students who know Tamil, French & Hindi.

GIVEN :
n(U) = 4000
n(F) = 2000
n(T) = 3000
n(H) = 500
n( F ∩ T)= 1500
n( F ∩ H)= 300
n( T ∩ H)= 200
n( T ∩ H ∩ F)= 50

i) Number of students who do not know any of the three languages = n(U) - { n(F) + n(T) + n(H) - n( F ∩ T) - n( F ∩ H) - n( T ∩ H) +n( T ∩ H ∩ F)

= 4000 - {2000 + 2000 + 500 - (1500 + 300 + 200) + 50}
= 4000 - 3550
Number of students who do not know any of the three languages = 450

ii)Number of students who know at least one language = n(F) + n(T) + n(H) - n( F ∩ T) - n( F ∩ H) - n( T ∩ H) +n( T ∩ H ∩ F)

= 2000 + 2000 + 500 - (1500 + 300 + 200) + 50}
Number of students who know at least one language = 3550

iii)Number of students who know only two languages = n( F ∩ T) - n( F ∩ H) - n( T ∩ H) +n( T ∩ H ∩ F)

= 1500 + 300 + 200 + 50
Number of students who know only two languages = 1850

HOPE THIS WILL HELP YOU...

Answer 2

Hi ,

According to the problem given ,

Total students in the school = U = 4000

Number of students who know ,

i ) French = n( F ) = 2000

ii ) Tamil = n( T ) = 3000

iii ) Hindi = n( H ) = 500

iv ) French and Tamil = n( F n T ) = 1500

v ) French and Hindi = n ( F n H ) = 300

vi ) Tamil and Hindi = n ( T n H ) = 200

vii ) French and Tamil and Hindi = n( FnTnH)

= 50

We know that ,

n( FUTUH)=n(F)+n(T)+n(H)-n(FnT)-n(FnH)-n(TnH)+n(FnTnH )

n( FUTUH ) = 2000 + 3000 + 500 - 1500 - 300 - 200 + 50

= 3550

i ) Number of students who doesn't know

any of the 3 languages = U - n( FUTUH )

= 4000 - 3550

= 450----( 1 )

ii ) Number of students who knows

at least one language

= n ( FUTUH )

= 3550

iii ) Number students who knows only two

languages = 1450 + 350 + 50

= 1850

I hope this helps you.

: )