Question

In a survey of $25$ students, it was found that $15$ had Mathematics, $12$ had taken physics and $11$ had taken Chemistry, $5$ had taken Mathematics and Chemistry, $9$ had taken Mathematics and Physics, $4$ had taken Physics and Chemistry and $3$ had taken all the three subjects. Find the number of students that had
(i) Only Chemistry (ii) Physics and Chemistry but not mathematics
(iii) Only one of the subjects (iv) None of the subjects
(v) at least one of the three subjects

Answer 1

R.E.F image

(i) $11-5-4+3=5$

(3 removed twice)

(ii) $4-3=1$

(iii) only maths $=15-9-5+3$

$=4$

only phy $=12-9-4+3=2$

only chem $=5\Rightarrow Total=11$

(iv) Atleast one $=11+9+5+4-2\left(3\right)$

$=29-6=23$

$\Rightarrow$ None $=25-23=2$

(v) $23$