In a class of 140 students numbered 1 to 140 , all even numbered students opted mathematics course,

Solution:

Let $n(A)=$ number of students opted Mathematics $=70$,

$\mathrm{n}(\mathrm{B})=$ number of students opted Physics $=46$,

$\mathrm{n}(\mathrm{C})=$ number of students opted Chemistry

$=28$

$\mathrm{n}(\mathrm{A} \cap \mathrm{B})=23$

$\mathrm{n}(\mathrm{B} \cap \mathrm{C})=9$

$\mathrm{n}(\mathrm{A} \cap \mathrm{C})=14$

$\mathrm{n}(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C})=4$

Now $\mathrm{n}(\mathrm{A} \cup \mathrm{B} \cup \mathrm{C})$

$=n(A)+n(B)+n(C)-n(A \cap B)-n(B \cap C)$

$-n(A \cap C)+n(A \cap B \cap C)$

$=70+46+28-23-9-14+4=102$

So number of students not opted for any course

$=$ Total $-\mathrm{n}(\mathrm{A} \cup \mathrm{B} \cup \mathrm{C})$

$=140-102=38$