$A$ lady gives a dinner party for $6$ guests. The number of ways in which they may be selected from among $10$ friends,if two of the friends will not attend the party together,is:

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

$A$ student is required to answer $10$ questions out of $13$ in an examination,such that he must choose at least $4$ questions from the first $5$ questions. How many ways can he make the selection?

If $^{10}C_r = ^{10}C_{r+2}$,then $^5C_r$ equals

$A$ question paper is divided into two parts,$A$ and $B$,each consisting of $5$ questions. If a candidate has to select $6$ questions in total,in how many ways can they do so if they must select at least $2$ questions from each part?

$A$ committee of $11$ members is to be formed from $8$ males and $5$ females. If $m$ is the number of ways the committee is formed with at least $6$ males and $n$ is the number of ways the committee is formed with at least $3$ females,then: