Question

In a class of 52 students, 13 excel in Science and Mathematics, 16 excel in Science and Arts,

12 excel in Mathematics and Arts, 24 excel in Arts and 2 excel in none. Twice as many students
excel in science only as do in mathematics only. The number of students who excel in
mathematics only is six times the number of students who excel in Arts only. Determine the
number of students who excelled in:
(a) All the three subjects
(b) Two subjects. Use ven diagram also

Answer 1

The number of students who excelled in:

(a) All three subjects: 2

(b) Two subjects: 39

Let's denote the sets as follows:

S for the set of students excelling in Science

M for the set of students excelling in Mathematics

A for the set of students excelling in Arts

We are given the following information:

n(S∩M)=13

n(S∩A)=16

n(M∩A)=12

n(S∩M∩A)=2

n(none)=2

n(S∖(M∪A))=2 (Twice as many students excel in Science only as do in Mathematics only)

n(M∖(S∪A))=6 (The number of students who excel in Mathematics only is six times the number of students who excel in Arts only.)

Now, let's determine the number of students who excelled in each category:

(a) All three subjects:

n(S∩M∩A)=2

(b) Two subjects:

n(S∩M)+n(S∩A)+n(M∩A)−2n(S∩M∩A)=13+16+12−2(2)=39

So, the number of students who excelled in:

(a) All three subjects: 2

(b) Two subjects: 39