$20$ teachers of a school either teach mathematics or physics. $12$ of them teach mathematics while $4$ teach both the subjects. Then the number of teachers teaching physics is

Out of $800$ boys in a school,$224$ played cricket,$240$ played hockey,and $336$ played basketball. Of the total,$64$ played both basketball and hockey,$80$ played cricket and basketball,and $40$ played cricket and hockey. $24$ played all three games. The number of boys who did not play any game is:

The sum of all the elements of the set $\{\alpha \in \{1, 2, \ldots, 100\} : \operatorname{HCF}(\alpha, 24) = 1\}$ is

The lengths of the sides and the diagonal of an isosceles trapezium form a two-element set $\{a, b\}$. If $a > b$,then $a / b$ equals

Let $A$ denote the set of all $2$-digit numbers in base $10$ that are equal to four times the sum of the factorials of their digits. The sum of the numbers in $A$ is

The probability that at least one of the events $E_1$ and $E_2$ occurs is $0.6$. If the probability of the simultaneous occurrence of $E_1$ and $E_2$ is $0.2$,then $P(E_1') + P(E_2') = $