Question

In a survey of 25 students, it was found that 12 have taken physics, 11 have taken chemistry and 15 have taken mathematics; 4 have taken physics and chemistry; 9 have taken physics and mathematics; 5 have taken chemistry and mathematics while 3 have taken all the three subjects.
Find the number of students who have taken

(i) physics only;
(ii) chemistry only;
(iii) mathematics only;
(iv) physics and chemistry but not mathematics;
(v) physics and mathematics but not chemistry;
(vi) only one of the subjects;
(vii) at least one of the three subjects;
(viii) none of the three subjects.

Answer 1

Let P, C and Al be the sets of students who have taken physics, chemistry and mathematics respectively.
Let a, b, c, d, e, f andg denote the number of students in the respective regions, as shown in the adjoining Venn. diagram.

As per data given, we have
$\{\begin{array}{c}a+b+c+d=12\\ b+c+e+f=11\\ c+d+f+f+15\\ b+c=4\\ c+d=9\\ c+f=5\\ c=3\end{array}$

From these equations, we get
$c=3,f=2,d=6.b=1$.

Now, $c+d+f+g=15\Rightarrow 3+6+2+g=15\Rightarrow g=14;b+c+e+f=11\Rightarrow 1+3+e+2=11\Rightarrow e=5;a+b+c+d=12\Rightarrow a+1+3+6=12\Rightarrow a=2;\therefore a=2,b=1,c=3,d=6,e=5,f=2andg=4$
So, we have:

(i) Number of students who offered physics only = a= 2.

(ii) Number of students who offered chemistry only = e= 5.

(iii) Number of students who offered mathematics only g = 4.

(iv) Number of students who offered physics and chemistry but not mathematics = b =1.

(v) Number of students who offered physics and mathematics but not chemistry = d = 6.

(vi) Number of students who offered only one of the given subjects
= (a+e+g) = (2 +5+4) = 11.

(vii) Number of students who offered at least one of the given subjects=(a+b+c+d+c+f+g)=(2 +1 +3 +6+5 +2+4)= 23.

(viii) Number of students who offered none of the three given subjects = (25 - 23) = 2.