An eight-digit number divisible by $9$ is to be formed using digits from $0$ to $9$ without repeating the digits. The number of ways in which this can be done is: (in $(7!)$)

In how many ways can a committee,consisting of $5$ men and $6$ women,be formed from $8$ men and $10$ women?

$^{n - 1}C_r = (k^2 - 3) \cdot ^nC_{r + 1}$ if $k \in$

Given $6$ different letters of an alphabet. Words with $4$ letters are formed from these given letters. The number of words which have at least one letter repeated and no two same letters are together,is:

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?

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: