Everybody in a room shakes hands with everybody else. The total number of handshakes is $66$. The total number of persons in the room is

How many words of $4$ consonants and $3$ vowels can be formed from $6$ consonants and $5$ vowels?

In an election,there are $8$ candidates,out of which $5$ are to be chosen. If a voter may vote for any number of candidates but not greater than the number to be chosen,then in how many ways can a voter vote?

If the number of $5$-element subsets of the set $A = \{a_1, a_2, \dots, a_{20}\}$ containing $20$ distinct elements is $k$ times the number of $5$-element subsets containing the element $a_4$,then what is the value of $k$?

If a committee of $10$ members is to be formed from $8$ men and $6$ women,then the number of different possible committees in which the men are in majority is

If ${}^nC_{r-1}=36$,${}^nC_r=84$,and ${}^nC_{r+1}=126$,then the value of $nr^2$ is