# Out of 100 students; 15 passed in English, 12 passed in Mathematics, 8 in Science, 6 in English and Mathematics, 7 in Mathematics and Science ( ; 4 ) in English and Science ( ; 4 ) in all the three. Find

how many passed

(i) in-English and Mathematics but not in Science

(ii) in Mathematics and Science but not in English

(iii) in Mathematics only

(iv) in more than one subject only

Let ( mathrm{M} ) be the set of students who passed in Mathematics, ( mathrm{E} ) be the set of

students who passed in English and S be the set of students who passed

in Science.

Given ( n(U)=100 )

( n(E)=15, n(M)=12, n(S)=8 )

( mathrm{n}(mathrm{E} cap mathrm{M})=6, mathrm{n}(mathrm{M} cap mathrm{S})=7, mathrm{n}(mathrm{E} cap mathrm{S})-4, ) and ( mathrm{n}(mathrm{E} cap mathrm{M} cap mathrm{S})=4 )

From the figure, we have

[

begin{array}{ll}

a=n(E cap M cap S)=4

a+d=n(M cap S)=7

therefore=3

a+b=n(M cap E)=6

therefore quad b=2

a+c=n(S cap E)=4

therefore quad c=0

a+b+d+e=n(M)=12

therefore quad 4+2+3+e=12

therefore=3

end{array}

]

( a+b+c+g=15 )

( therefore quad 4+2+0+g=n(E)=15 )

( therefore quad g=9 )

( a+c+d+f=n(S)=8 )

( therefore quad 4+0+3+f=8 )

( therefore quad f=1 )

Number of students passed in English and Mathematics but not in

Science ( =b=2 )

(ii) Number of students passed in Mathematics and Science but not in

English ( =d=3 )

(iii) Number of students passed in Mathematics only ( =mathrm{e}=3 )

(iv) Number of students passed in more than one subject ( =a+b+c+d )

( =4+2+0+3=9 )