Question

Subject wise result of 30 students of a class is mentioned as follows: 15 students passed in English, 12 students passed in Mathematics, 8 students passed in science, 6 students passed in English and Mathematics, 7 students passed in Mathematics and science, 4 students passed in English| and Science, 4 students passed in all three subjects.
On the basis of above information, answer the following:
Then number of students who passed in English and Mathematics but not in science,
(a) 5 (b) 3
(c) 2
(d) none of these.
Then number of students who passed in Science and Mathematics but not in English (a) 3 (b)2
(c) 4
(d) none of these.
Then number of students who passed in Mathematics only. (a) 4 (b) 3
(c) 5
(d) none of these.
(a) 7
Then number of students who passed in more than one subject. (b) 8
(c) 9
(d) none of these
Then number of students who passed in none of the three subjects. (a) 8 (b) 10
(c) 9
(d) none of these.

Answer 1

Given:

No. of students in the class = 30

No. of students who passed in English = 15

No. of students who passed in Mathematics = 12

No. of students who passed in Science = 8

No. of students who passed in English and Mathematics = 6

No. of students who passed in Mathematics and Science = 7

No. of students who passed in English and Science = 4

No. of students who passed in all three subjects = 4

To find:

No. of students who passed in English and Mathematics but not in Science.

No. of students who passed in Science and Mathematics but not in English.

No. of students who passed in Mathematics only.

No. of students who passed in more than one subject.

No. of students who passed in none of the three subjects.

Solution:

Let M be those students who passed in Mathematics, E be those students who passed in English, S be those students who passed in Science.

Given that $n(U)=30, n(E)=15, n(M)=12,n(S)=8, n(E\cap M)=6,n(M\cap S)=7, n(E\cap S)=4, n(E\cap M\cap S)=4$

From the figure, it can be seen that,

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

Also,

$a+d=n(E\cap M)=6$

$4+d=6$

$d=6-4$

∴ $d=2$

Hence, no. of students who passed in English and Mathematics but not in Science is 2. Option (c) is the correct answer.

From figure,

$a+b=n(M\cap S)=7$

$4+b=7$

$b=7-4$

∴ $b=3$

Hence, no. of students who passed in Science and Mathematics but not in English is 3. Option (a) is the correct answer.

From figure,

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

$4+3+2+e=12$

$9+e=12$

$e=12-9$

∴ $e=3$

Hence, no. of students who passed in Mathematics only is 3. Option (b) is the correct answer.

From figure,

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

$4+c=4$

$c=4-4$

∴ $c=0$

No. of students who passed in more than one subject

$=a+b+c+d$

$=4+3+0+2$

$=9$

9 students passed in more than one subject. Option (c) is the correct answer.

From figure,

$a+c+d+f=n(E)=15$

$4+0+2+f=15$

$6+f=15$

$f=15-6$

∴ $f=9$

From figure,

$a+b+c+g=n(S)=8$

$4+3+0+g=8$

$7+g=8$

$g=8-7$

∴ $g=1$

No. of students who passed one or more subjects

$=a+b+c+d+e+f+g$

$=4+3+0+2+3+9+1$

$=22$

No. of students who passed in none of the three subjects = No. of students in class - No. of students who passed one or more subjects

No. of students who passed in none of the three subjects = $30-22$

No. of students who passed in none of the three subjects = $8$

Hence, 8 students passed in none of the three subjects. Option (a) is the correct answer.

2 students passed in English and Mathematics but not in Science. Option (c) is the correct answer.

3 students passed in Science and Mathematics but not in English. Option (a) is the correct answer.

3 students passed in Mathematics only. Option (b) is the correct answer.

9 students passed in more than one subject. Option (c) is the correct answer.

8 students passed in none of the three subjects. Option (a) is the correct answer.