An organization awarded $48$ medals in event $A$,$25$ in event $B$,and $18$ in event $C$. If these medals were awarded to a total of $60$ men and only $5$ men received medals in all three events,how many men received medals in exactly two of the three events?

Let $A, B, C$ be three non-void subsets of set $S$. Let $(A \cap C) \cup (B \cap C^{\prime}) = \phi$, where $C^{\prime}$ denotes the complement of set $C$ in $S$. Then:

Let $A = \{1, 2, 3, 4, 5, 6, 7\}$ and $B = \{3, 6, 7, 9\}$. Then the number of elements in the set $\{ C \subseteq A : C \cap B \neq \phi \}$ is

If a set $A$ has $2n + 1$ elements,then how many subsets of $A$ contain at least $n$ elements?

In a school of $800$ boys,$224$ play cricket,$240$ play hockey,and $336$ play basketball. Of the total,$64$ play basketball and hockey,$80$ play cricket and basketball,and $40$ play cricket and hockey,while $24$ play all three games. Find the number of boys who do not play any game.

Let $S = \{1, 2, 3, 4, 5, 6, 9\}$. Then the number of elements in the set $T = \{A \subseteq S : A \neq \phi \text{ and the sum of all the elements of } A \text{ is not a multiple of } 3\}$ is ..... .