Everybody in a room shakes hand with everybody else. The total number of hand shakes is $66$. The total number of persons in the room is
$11$
$12$
$13$
$14$
If $^{2n}{C_2}{:^n}{C_2} = 9:2$ and $^n{C_r} = 10$, then $r = $
Determine the number of $5 -$ card combinations out of a deck of $52$ cards if each selection of $5$ cards has exactly one king.
There are $5$ students in class $10,6$ students in class $11$ and $8$ students in class $12.$ If the number of ways, in which $10$ students can be selected from them so as to include at least $2$ students from each class and at most $5$ students from the total $11$ students of class $10$ and $11$ is $100 \mathrm{k}$, then $\mathrm{k}$ is equal to $......$
$^n{P_r}{ \div ^n}{C_r}$ =
A set contains $2n + 1$ elements. The number of subsets of this set containing more than $n$ elements is equal to