$A$ bag contains $30$ balls numbered from $1$ to $30$. One ball is drawn randomly. The probability that the number on the ball is a multiple of $5$ or $7$ is:

If $X$ and $Y$ are two sets such that $n(X) = 17$,$n(Y) = 23$ and $n(X \cup Y) = 38$,find $n(X \cap Y)$.

In a battle,$70\%$ of the combatants lost one eye,$80\%$ an ear,$75\%$ an arm,$85\%$ a leg,and $x\%$ lost all the four limbs. The minimum value of $x$ is:

If $A = \{1, 2, 3, 4, 5, 6\}$,then the number of subsets of $A$ which contain at least two elements is

Let $A_1, A_2, \ldots, A_m$ be non-empty subsets of $\{1, 2, 3, \ldots, 100\}$ satisfying the following conditions:
$1.$ The numbers $|A_1|, |A_2|, \ldots, |A_m|$ are distinct.
$2.$ $A_1, A_2, \ldots, A_m$ are pairwise disjoint.
(Here $|A|$ denotes the number of elements in the set $A$).
Then,the maximum possible value of $m$ is:

Let $X$ be a set of $5$ elements. The number $d$ of ordered pairs $(A, B)$ of subsets of $X$ such that $A \neq \phi, B \neq \phi, A \cap B \neq \phi$ satisfies: