Let $X = \{n \in N : 1 \leq n \leq 50\}$. If $A = \{n \in X : n \text{ is a multiple of } 2\}$ and $B = \{n \in X : n \text{ is a multiple of } 7\}$,then the number of elements in the smallest subset of $X$ containing both $A$ and $B$ is

There are $200$ individuals with a skin disorder,$120$ had been exposed to the chemical $C_{1}$,$50$ to chemical $C_{2}$,and $30$ to both the chemicals $C_{1}$ and $C_{2}$. Find the number of individuals exposed to chemical $C_{2}$ but not chemical $C_{1}$.

$205$ students take an examination,of whom $105$ pass in English,$70$ students pass in Mathematics,and $30$ students pass in both. How many students fail in both subjects?

$A$ college awarded $38$ medals in football,$15$ in basketball and $20$ in cricket. If these medals went to a total of $58$ men and only three men got medals in all the three sports,how many received medals in exactly two of the three sports?

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

In a class of $55$ students,$23$ study Mathematics,$24$ study Physics,$19$ study Chemistry,$12$ study Mathematics and Physics,$9$ study Mathematics and Chemistry,$7$ study Physics and Chemistry,and $4$ study all three subjects. Find the number of students who study exactly one subject.